3.1 Upwind schemes

In the classical approach dating back to the original Godunov scheme (Reference Godunov1959), the space–time control volume is selected to be $C_{j}\times [t^{n},t^{n+1}]$ . This results in

(3.2) $$\begin{eqnarray}\overline{\boldsymbol{U}}_{j}^{n+1}=\overline{\boldsymbol{U}}_{j}^{n}-\displaystyle \frac{\unicode[STIX]{x1D6E5}t^{n}}{\unicode[STIX]{x1D6E5}x}\Bigl[\boldsymbol{{\mathcal{F}}}_{j+1/2}^{\,n+1/2}-\boldsymbol{{\mathcal{F}}}_{j-1/2}^{\,n+1/2}\Bigr],\end{eqnarray}$$

where

(3.3) $$\begin{eqnarray}\boldsymbol{{\mathcal{F}}}_{j+1/2}^{\,n+1/2}\approx \displaystyle \frac{1}{\unicode[STIX]{x1D6E5}t^{n}}\int _{t^{n}}^{t^{n+1}}\boldsymbol{F}(\boldsymbol{U}(x_{j+1/2},t))\,\text{d}t\end{eqnarray}$$

is a numerical flux, which needs to be computed based on the data (3.1) available at time level $t=t^{n}$ . In order to evaluate the time integral in (3.3), one has to (approximately) solve the following initial value problem (IVP) with the initial data prescribed at time $t=t^{n}$ :

(3.4) $$\begin{eqnarray}\displaystyle \boldsymbol{U}_{t}+\boldsymbol{F}(\boldsymbol{U})_{x} & = & \displaystyle \mathbf{0},\quad t\in (t^{n},t^{n+1}],\nonumber\\ \displaystyle \boldsymbol{U}(x,t^{n}) & = & \displaystyle \left\{\begin{array}{@{}ll@{}}\boldsymbol{{\mathcal{P}}}_{j}^{n}(x),\quad & x<x_{j+1/2},\\ \boldsymbol{{\mathcal{P}}}_{j+1}^{n}(x),\quad & x>x_{j+1/2}.\end{array}\right.\end{eqnarray}$$

In the case of the first-order piecewise constant reconstruction, that is, when $\boldsymbol{{\mathcal{P}}}_{j}^{n}(x)=\,\overline{\boldsymbol{U}}_{j}^{n}$ for all $j$ , the IVP (3.4) is a Riemann problem, whose solution, as is well known, is self-similar:

$$\begin{eqnarray}\boldsymbol{U}(x,t)=\boldsymbol{U}_{j+1/2}^{n+}(\unicode[STIX]{x1D709}),\quad \unicode[STIX]{x1D709}:=\displaystyle \frac{x-x_{j+1/2}}{t-t^{n}}.\end{eqnarray}$$

Therefore it is constant at $x=x_{j+1/2}$ for all $t\in (t^{n},t^{n+1}]$ and the numerical flux (3.3) becomes

$$\begin{eqnarray}\boldsymbol{{\mathcal{F}}}_{j+1/2}^{\,n+1/2}=\boldsymbol{F}(\boldsymbol{U}_{j+1/2}^{n+}(0)).\end{eqnarray}$$

In order to complete the construction of the first-order Godunov scheme one then needs to analytically solve the Riemann problem (3.4) to find $\boldsymbol{U}_{j+1/2}^{n+}(0)$ . For some hyperbolic systems of conservation laws this can be done; see, for example, Godlewski and Raviart (Reference Godlewski and Raviart1996), Kröner (Reference Kröner1997) and LeVeque (Reference LeVeque2002). Alternatively, the Riemann problem (3.4) can be solved approximately; a variety of approximate Riemann problem solvers can be found in Godlewski and Raviart (Reference Godlewski and Raviart1996), Kröner (Reference Kröner1997), LeVeque (Reference LeVeque2002) and Toro (Reference Toro2009), for example.

If (3.1) is a piecewise linear function used in the construction of second-order upwind schemes, the IVP (3.4) is a generalized Riemann problem (GRP), whose solution is no longer self-similar. GRPs are much harder to solve analytically. Although this was done for some hyperbolic systems of conservation laws in Ben-Artzi (Reference Ben-Artzi1989), Ben-Artzi and Falcovitz (Reference Ben-Artzi and Falcovitz1984, Reference Ben-Artzi and Falcovitz1986, Reference Ben-Artzi and Falcovitz2003) and Ben-Artzi, Li and Warnecke (Reference Ben-Artzi, Li and Warnecke2006), approximate GRP solvers are more popular as they are much easier to construct and implement; see, for example, Godlewski and Raviart (Reference Godlewski and Raviart1996), LeVeque (Reference LeVeque2002) and Lukáčová-Medvid’ová, Morton and Warnecke (Reference Lukáčová-Medvid’ová, Morton and Warnecke2004).

A simpler and thus more popular approach for constructing high-order upwind schemes is by using the semi-discrete formulation of (2.2), which is obtained by integrating (2.2) over the cell $C_{j}$ , that is,

(3.5) $$\begin{eqnarray}\displaystyle \frac{\text{d}}{\text{d}t}\,\overline{\boldsymbol{U}}_{j}(t)=-\displaystyle \frac{1}{\unicode[STIX]{x1D6E5}x}[\boldsymbol{{\mathcal{F}}}_{j+1/2}(t)-\boldsymbol{{\mathcal{F}}}_{j-1/2}(t)],\end{eqnarray}$$

where

$$\begin{eqnarray}\overline{\boldsymbol{U}}_{j}(t)\approx \displaystyle \frac{1}{\unicode[STIX]{x1D6E5}x}\int _{C_{j}}\boldsymbol{U}(x,t)\,\text{d}x\end{eqnarray}$$

are cell averages assumed to be available at a certain time level $t$ and the numerical fluxes $\boldsymbol{{\mathcal{F}}}_{j+1/2}(t)$ are obtained by (approximately) solving the following Riemann problem:

$$\begin{eqnarray}\displaystyle \boldsymbol{U}_{t}+\boldsymbol{F}(\boldsymbol{U})_{x} & = & \displaystyle \mathbf{0},\quad t\in (t,t+\unicode[STIX]{x1D70F}],\nonumber\\ \displaystyle \boldsymbol{U}(x,t) & = & \displaystyle \left\{\begin{array}{@{}ll@{}}\boldsymbol{U}_{j+1/2}^{-}(t),\quad & ~~x<x_{j+1/2},\\ \boldsymbol{U}_{j+1/2}^{+}(t),\quad & ~~x>x_{j+1/2}.\end{array}\right.\nonumber\end{eqnarray}$$

Here, $\unicode[STIX]{x1D70F}$ is a small positive number and $\boldsymbol{U}_{j+1/2}^{+}(t)$ and $\boldsymbol{U}_{j+1/2}^{-}(t)$ are the right- and left-sided values of the piecewise polynomial interpolant (reconstructed at time $t$ from the cell averages $\overline{\boldsymbol{U}}_{j}(t)$ )

(3.6) $$\begin{eqnarray}\widetilde{\boldsymbol{U}}(x;t)=\mathop{\sum }_{j}\boldsymbol{{\mathcal{P}}}_{j}(x;t)\unicode[STIX]{x1D712}_{j}(x),\end{eqnarray}$$

namely,

(3.7) $$\begin{eqnarray}\boldsymbol{U}_{j+1/2}^{+}(t)=\boldsymbol{{\mathcal{P}}}_{j+1}(x_{j+1/2};t)\quad \text{and}\quad \boldsymbol{U}_{j+1/2}^{-}(t)=\boldsymbol{{\mathcal{P}}}_{j}(x_{j+1/2};t).\end{eqnarray}$$

The semi-discrete scheme is implemented by numerically solving the ODE system (3.5) using an appropriate ODE solver. A popular choice of such solvers is that of strong stability preserving (SSP) Runge–Kutta and multistage methods; see, for example, Gottlieb, Ketcheson and Shu (Reference Gottlieb, Ketcheson and Shu2011) and Gottlieb, Shu and Tadmor (Reference Gottlieb, Shu and Tadmor2001). These methods were originally designed to ensure that the total variation of the computed solution does not increase at the time evolution stage and thus they were originally referred to as total-variation-diminishing (TVD) methods in Shu (Reference Shu1988) and Shu and Osher (Reference Shu and Osher1988).

3.2 Central schemes

Central schemes offer a simpler, Riemann-problem-solver-free alternative to upwind schemes. They are obtained by selecting shifted space–time control volumes that contain the corresponding Riemann fans. The simplest choice of such control volumes, $[x_{j},x_{j+1}]\times [t^{n},t^{n+1}]$ , was suggested in Nessyahu and Tadmor (Reference Nessyahu and Tadmor1990), where a second-order staggered central scheme (the Nessyahu–Tadmor scheme) was derived. The cell averages at time level $t=t^{n+1}$ are then obtained over the staggered grid (compare with (3.2)):

(3.8) $$\begin{eqnarray}\overline{\boldsymbol{U}}_{j+1/2}^{n+1}=\displaystyle \frac{1}{\unicode[STIX]{x1D6E5}x}\int _{x_{j}}^{x_{j+1}}\widetilde{\boldsymbol{U}}^{n}(x)\,\text{d}x-\displaystyle \frac{\unicode[STIX]{x1D6E5}t^{n}}{\unicode[STIX]{x1D6E5}x}[\boldsymbol{{\mathcal{F}}}_{j+1}^{\,n+1/2}-\boldsymbol{{\mathcal{F}}}_{j}^{\,n+1/2}],\end{eqnarray}$$

where the numerical fluxes are

(3.9) $$\begin{eqnarray}\boldsymbol{{\mathcal{F}}}_{j}^{\,n+1/2}\approx \displaystyle \frac{1}{\unicode[STIX]{x1D6E5}t^{n}}\int _{t^{n}}^{t^{n+1}}\boldsymbol{F}(\boldsymbol{U}(x_{j},t))\,\text{d}t.\end{eqnarray}$$

We note that the spatial integral on the right-hand side of (3.8) can be explicitly computed for any piecewise polynomial reconstruction $\widetilde{\boldsymbol{U}}^{n}$ . Further, the time integrals in (3.9) can be easily computed using an appropriate quadrature because the solution of the IVP

$$\begin{eqnarray}\displaystyle \boldsymbol{U}_{t}+\boldsymbol{F}(\boldsymbol{U})_{x} & = & \displaystyle \mathbf{0},\quad t\in (t^{n},t^{n+1}],\nonumber\\ \displaystyle \boldsymbol{U}(x,t^{n}) & = & \displaystyle \widetilde{\boldsymbol{U}}^{n}(x)\nonumber\end{eqnarray}$$

remains smooth at the cell centres $x=x_{j}$ , provided the CFL number is taken to be less than or equal to 1/2, in order to prevent the nonlinear (possibly discontinuous) waves generated at cell interfaces at time level $t=t^{n}$ from reaching the cell centres before $t=t^{n+1}$ .

The second-order Nessyahu–Tadmor scheme is designed by taking the second-order piecewise linear reconstruction, for which

$$\begin{eqnarray}\boldsymbol{{\mathcal{P}}}_{j}^{n}(x)=\,\overline{\boldsymbol{U}}_{j}^{n}+(\boldsymbol{U}_{x})_{j}^{n}(x-x_{j}),\end{eqnarray}$$

where $(\boldsymbol{U}_{x})_{j}^{n}$ are the slopes that approximate $\boldsymbol{U}_{x}(x_{j},t^{n})$ in a non-oscillatory manner using a nonlinear slope limiter, and the midpoint quadrature for the temporal integrals in (3.9). This results in

(3.10) $$\begin{eqnarray}\displaystyle \overline{\boldsymbol{U}}_{j+1/2}^{n+1} & = & \displaystyle \displaystyle \frac{1}{2}(\,\overline{\boldsymbol{U}}_{j}^{n}+\,\overline{\boldsymbol{U}}_{j+1}^{n})+\displaystyle \frac{\unicode[STIX]{x1D6E5}x}{8}[(\boldsymbol{U}_{x})_{j}^{n}-(\boldsymbol{U}_{x})_{j+1}^{n}]\nonumber\\ \displaystyle & & \displaystyle \qquad -\displaystyle \frac{\unicode[STIX]{x1D6E5}t^{n}}{\unicode[STIX]{x1D6E5}x}[\boldsymbol{F}(\boldsymbol{U}_{j+1}^{n+1/2})-\boldsymbol{F}(\boldsymbol{U}_{j}^{n+1/2})],\end{eqnarray}$$

where the midpoint values $\boldsymbol{U}_{j}^{n+1/2}$ are predicted using the Taylor expansion as follows:

(3.11) $$\begin{eqnarray}\boldsymbol{U}_{j}^{n+1/2}=\,\overline{\boldsymbol{U}}_{j}^{n}+\displaystyle \frac{\unicode[STIX]{x1D6E5}t}{2}(\boldsymbol{U}_{t})_{j}^{n}=\,\overline{\boldsymbol{U}}_{j}^{n}-\displaystyle \frac{\unicode[STIX]{x1D6E5}t}{2}(\boldsymbol{F}(\boldsymbol{U})_{x})_{j}^{n}.\end{eqnarray}$$

Here, the derivatives $(\boldsymbol{F}(\boldsymbol{U})_{x})_{j}^{n}$ can be computed with the help of the same limiter, which was used in the computation of $(\boldsymbol{U}_{x})_{j}^{n}$ .

Remark 3.3. Note that if all of the slopes $(\boldsymbol{U}_{x})_{j}^{n}$ and $(\boldsymbol{F}(\boldsymbol{U})_{x})_{j}^{n}$ are set to zero, the Nessyahu–Tadmor scheme (3.10), (3.11) will reduce to the first-order staggered Lax–Friedrichs scheme

(3.12) $$\begin{eqnarray}\overline{\boldsymbol{U}}_{j+1/2}^{n+1}=\displaystyle \frac{1}{2}(\,\overline{\boldsymbol{U}}_{j}^{n}+\,\overline{\boldsymbol{U}}_{j+1}^{n})-\displaystyle \frac{\unicode[STIX]{x1D6E5}t^{n}}{\unicode[STIX]{x1D6E5}x}[\boldsymbol{F}(\,\overline{\boldsymbol{U}}_{j+1}^{n})-\boldsymbol{F}(\,\overline{\boldsymbol{U}}_{j}^{n})],\end{eqnarray}$$

which is a Godunov-type version of the classical Lax–Friedrichs scheme from Friedrichs (Reference Friedrichs1954) and Lax (Reference Lax1954).

Remark 3.5. The staggered central schemes provide a ‘black-box’ tool for solving general (multidimensional) hyperbolic systems of conservation laws. However, the amount of numerical diffusion present in staggered central schemes is quite large and it significantly increases when the time step is taken to be small or, alternatively, when a long-term (steady-state) computation is performed. In order to clarify this point, let us consider the first-order staggered Lax–Friedrichs scheme (3.12) and rewrite it in the following equivalent form:

(3.13) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \frac{\overline{\boldsymbol{U}}_{j+1/2}^{n+1}-\overline{\boldsymbol{U}}_{j+1/2}^{n}}{\unicode[STIX]{x1D6E5}t^{n}}+\displaystyle \frac{\boldsymbol{F}(\,\overline{\boldsymbol{U}}_{j+1}^{n})-\boldsymbol{F}(\,\overline{\boldsymbol{U}}_{j}^{n})}{\unicode[STIX]{x1D6E5}x}=\displaystyle \frac{(\unicode[STIX]{x1D6E5}x)^{2}}{8\unicode[STIX]{x1D6E5}t^{n}}\cdot \displaystyle \frac{\overline{\boldsymbol{U}}_{j+1}^{n}-2\,\overline{\boldsymbol{U}}_{j+1/2}^{n}+\overline{\boldsymbol{U}}_{j}^{n}}{(\unicode[STIX]{x1D6E5}x/2)^{2}}.\end{eqnarray}$$

As one can see, the terms on the left-hand side of (3.13) approximate $\boldsymbol{U}_{t}$ and $\boldsymbol{F}(\boldsymbol{U})_{x}$ , respectively, while the term on the right-hand side of (3.13) represents the numerical viscosity/diffusion present in the scheme. Notice that the numerical viscosity coefficient is proportional to $(\unicode[STIX]{x1D6E5}x)^{2}/\unicode[STIX]{x1D6E5}t^{n}$ and its size depends on the ratio of $\unicode[STIX]{x1D6E5}t^{n}$ and $\unicode[STIX]{x1D6E5}x$ . In the hyperbolic regime, one typically takes $\unicode[STIX]{x1D6E5}t^{n}\sim \unicode[STIX]{x1D6E5}x$ , and then the viscosity coefficient is proportional to $\unicode[STIX]{x1D6E5}x$ as it is supposed to be for first-order schemes. If, however, $\unicode[STIX]{x1D6E5}t^{n}$ is for some reason taken to be small then the numerical dissipation increases so significantly that the scheme may become inconsistent (e.g. if $\unicode[STIX]{x1D6E5}t^{n}\sim (\unicode[STIX]{x1D6E5}x)^{2}$ ). In addition, the excessive numerical dissipation present in staggered central schemes may badly affect the quality of solutions computed at large times (this makes the schemes inappropriate for capturing time-independent steady-state solutions, as pointed out in Kurganov and Tadmor Reference Kurganov and Tadmor2000).

In order to overcome this drawback, we have developed a new class of Riemann-problem-solver-free methods: central-upwind schemes, which are briefly described in Section 3.3. We refer the reader to Kurganov (Reference Kurganov2016), Kurganov and Lin (Reference Kurganov and Lin2007) and Kurganov et al. (Reference Kurganov, Prugger and Wu2017) for details of their derivation.

3.3 Central-upwind schemes

Central-upwind schemes are derived using the Godunov-type central approach described in Section 3.2, but using a different set of non-uniform space–time control volumes whose size is determined based on the one-sided local speeds of propagation $a_{j+1/2}^{\pm }$ , which are determined as follows. We first note that the piecewise polynomial reconstruction (3.6) is generically discontinuous at the cell interfaces $x=x_{j+1/2}$ and the (non-smooth) waves generated there propagate with local speeds whose upper and lower bounds can be estimated using the largest and smallest eigenvalues of the Jacobian $\unicode[STIX]{x2202}\boldsymbol{F}/\unicode[STIX]{x2202}\boldsymbol{U}$ . In most cases, reliable estimates are given by

(3.14a ) $$\begin{eqnarray}\displaystyle a_{j+1/2}^{+}(t) & = & \displaystyle \max \biggl\{\unicode[STIX]{x1D706}_{N}\biggl(\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{F}}{\unicode[STIX]{x2202}\boldsymbol{U}}(\boldsymbol{U}_{j+1/2}^{-}(t))\biggr),\unicode[STIX]{x1D706}_{N}\biggl(\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{F}}{\unicode[STIX]{x2202}\boldsymbol{U}}(\boldsymbol{U}_{j+1/2}^{+}(t))\biggr),\,0\biggr\},\end{eqnarray}$$

(3.14b ) $$\begin{eqnarray}\displaystyle a_{j+1/2}^{-}(t) & = & \displaystyle \min \biggl\{\unicode[STIX]{x1D706}_{1}\biggl(\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{F}}{\unicode[STIX]{x2202}\boldsymbol{U}}(\boldsymbol{U}_{j+1/2}^{-}(t))\biggr),\,\unicode[STIX]{x1D706}_{1}\biggl(\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{F}}{\unicode[STIX]{x2202}\boldsymbol{U}}(\boldsymbol{U}_{j+1/2}^{+}(t))\biggr),\,0\biggr\}.\end{eqnarray}$$

$\boldsymbol{F}$

Equipped with the set of non-uniform space–time control volumes, the solution is evolved to the new time level at which it is given as a set of cell averages of $\boldsymbol{U}$ over a new (strictly) non-uniform mesh containing twice as many cells as the original (uniform) mesh. The obtained solution is then projected onto the original grid, which makes the resulting fully discrete central-upwind scheme practically feasible; see Kurganov (Reference Kurganov2016) and Kurganov and Lin (Reference Kurganov and Lin2007) for details of their derivation. The fully discrete central-upwind scheme is, however, quite complicated (especially its 2-D version recently developed in Kurganov et al. Reference Kurganov, Prugger and Wu2017).

The central-upwind schemes may be significantly simplified if one passes to a semi-discrete limit by taking $\max _{n}(\unicode[STIX]{x1D6E5}t^{n})\rightarrow 0$ . This leads to a class of semi-discrete central-upwind schemes developed in Kurganov and Lin (Reference Kurganov and Lin2007), Kurganov et al. (Reference Kurganov, Noelle and Petrova2001) and Kurganov and Tadmor (Reference Kurganov and Tadmor2000), which in the 1-D case can be put into the flux form (3.5) with the central-upwind numerical flux

(3.15) $$\begin{eqnarray}\displaystyle \boldsymbol{{\mathcal{F}}}_{j+1/2} & = & \displaystyle \displaystyle \frac{a_{j+1/2}^{+}\boldsymbol{F}(\boldsymbol{U}_{j+1/2}^{-})-a_{j+1/2}^{-}\boldsymbol{F}(\boldsymbol{U}_{j+1/2}^{+})}{a_{j+1/2}^{+}-a_{j+1/2}^{-}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\displaystyle \frac{a_{j+1/2}^{+}a_{j+1/2}^{-}}{a_{j+1/2}^{+}-a_{j+1/2}^{-}}[\boldsymbol{U}_{j+1/2}^{+}-\boldsymbol{U}_{j+1/2}^{-}-\unicode[STIX]{x1D6FF}\,\boldsymbol{U}_{j+1/2}].\end{eqnarray}$$

Here,

(3.16) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\,\boldsymbol{U}_{j+1/2}:=\operatorname{minmod}(\boldsymbol{U}_{j+1/2}^{+}-\boldsymbol{U}_{j+1/2}^{\ast },\,\boldsymbol{U}_{j+1/2}^{\ast }-\boldsymbol{U}_{j+1/2}^{-})\end{eqnarray}$$

is the built-in anti-diffusion term (see Kurganov Reference Kurganov2016, Kurganov and Lin Reference Kurganov and Lin2007 for details of its derivation) and

(3.17) $$\begin{eqnarray}\displaystyle \boldsymbol{U}_{j+1/2}^{\ast }=\displaystyle \frac{a_{j+1/2}^{+}\boldsymbol{U}_{j+1/2}^{+}-a_{j+1/2}^{-}\boldsymbol{U}_{j+1/2}^{-}-\{\boldsymbol{F}(\boldsymbol{U}_{j+1/2}^{+})-\boldsymbol{F}(\boldsymbol{U}_{j+1/2}^{-})\}}{a_{j+1/2}^{+}-a_{j+1/2}^{-}} & & \displaystyle\end{eqnarray}$$

are the intermediate values. Finally, the $\operatorname{minmod}$ function in (3.16) is defined as follows:

$$\begin{eqnarray}\operatorname{minmod}(z_{1},z_{2},\ldots )=\left\{\begin{array}{@{}ll@{}}\min (z_{1},z_{2},\ldots ),\quad & \text{if}~z_{i}>0,\text{for all}~i,\\ \max (z_{1},z_{2},\ldots ),\quad & \text{if}~z_{i}<0,\text{for all}~i,\\ 0,\quad & \text{otherwise}.\end{array}\right.\end{eqnarray}$$

Note that all of the indexed quantities in (3.15)–(3.17) are time-dependent, but we omit this dependence for the sake of brevity.

5.1 Central-upwind schemes for the one-dimensional Saint-Venant system

We first consider the 1-D Saint-Venant system (2.3), which is the 1-D system of balance laws (2.1) with

$$\begin{eqnarray}\boldsymbol{U}=(h,q)^{\top },\quad \boldsymbol{F}(h,q)=\biggl(q,\displaystyle \frac{q^{2}}{h}+\displaystyle \frac{gh^{2}}{2}\biggr)^{\top }\quad \text{and}\quad \boldsymbol{S}(h;B)=(0,-ghB_{x})^{\top }.\end{eqnarray}$$

A semi-discrete scheme for the system of balance laws (2.2) reads as

(5.1) $$\begin{eqnarray}\displaystyle \displaystyle \frac{\text{d}}{\text{d}t}\,\overline{\boldsymbol{U}}_{j}=-\displaystyle \frac{1}{\unicode[STIX]{x1D6E5}x}[\boldsymbol{{\mathcal{F}}}_{j+1/2}-\boldsymbol{{\mathcal{F}}}_{j-1/2}]+\,\overline{\boldsymbol{S}}_{j}, & & \displaystyle\end{eqnarray}$$

where

(5.2) $$\begin{eqnarray}\overline{\boldsymbol{S}}_{j}\approx \displaystyle \frac{1}{\unicode[STIX]{x1D6E5}x}\int _{C_{j}}\boldsymbol{S}(\boldsymbol{U})\,\text{d}x\end{eqnarray}$$

is an approximation of the source term cell average.

When the semi-discrete central-upwind scheme for the system of conservation laws (2.2) is extended to the system of balance laws (2.1), the fluxes are still given by (3.15)–(3.17) and one only needs to select an appropriate quadrature for the integral on the right-hand side of (5.2). This seems to be quite easy, but the problem here is that the use of a standard quadrature, for example the midpoint rule that results in

(5.3) $$\begin{eqnarray}\overline{S}_{j}^{(2)}=-ghB_{x}(x_{j}),\end{eqnarray}$$

leads to a non-well-balanced scheme, which does not respect the balance between the flux and source terms in (2.3). As demonstrated in Kurganov and Levy (Reference Kurganov and Levy2002, Example 1), the resulting scheme (5.1)–(5.3), (3.15)–(3.17) is incapable of exactly preserving steady-state solutions of (2.3) and, as a result, it fails to accurately capture small perturbations of steady-state (quasi-steady-state) solutions, as the truncation error of the scheme may be larger than the magnitude of the waves to be captured.

In general, a good well-balanced numerical scheme for the 1-D Saint-Venant system should be able to exactly preserve smooth steady-state solutions of (2.3), which are given by

(5.4) $$\begin{eqnarray}q\equiv \widehat{q}=\text{const.},\quad E:=\displaystyle \frac{u^{2}}{2}+g(h+B)\equiv \widehat{E}=\text{const.}\end{eqnarray}$$

In fact, the most important (from the practical point of view) steady state out of those in (5.4) is a so-called ‘lake at rest’ state,

(5.5) $$\begin{eqnarray}q\equiv 0,\quad w=h+B\equiv \widehat{w}=\text{const.},\end{eqnarray}$$

which corresponds to still water with a flat water surface. We note that in many applications the water waves that must be captured are, in fact, small perturbations of the ‘lake at rest’ steady state.

5.1.1 Still-water equilibria-preserving central-upwind scheme

In order to design a still-water equilibria-preserving central-upwind scheme, we follow the main ideas presented in Kurganov and Levy (Reference Kurganov and Levy2002) and Kurganov and Petrova (Reference Kurganov and Petrova2007) and proceed in the following way.

Step 1: piecewise linear reconstruction of the bottom topography. We first replace the original bottom topography function with its continuous piecewise linear interpolant consisting of the linear pieces that connect the points $(x_{j+1/2},B_{j+1/2})$ :

(5.6) $$\begin{eqnarray}\widetilde{B}(x)=B_{j-1/2}+(B_{j+1/2}-B_{j-1/2})\cdot \displaystyle \frac{x-x_{j-1/2}}{\unicode[STIX]{x1D6E5}x},\quad x\in C_{j},\end{eqnarray}$$

schematically shown in Figure 5.1. Here,

$$\begin{eqnarray}B_{j+1/2}:=\displaystyle \frac{B(x_{j+1/2}+0)+B(x_{j+1/2}-0)}{2},\end{eqnarray}$$

which reduces to $B_{j+1/2}=B(x_{j+1/2})$ if $B$ is continuous at $x=x_{j+1/2}$ .

Figure 5.1.

Bottom topography function $B$ and its piecewise linear approximation  $\widetilde{B}$ .

We note that since $\widetilde{B}$ is a piecewise linear function, its point value at $x=x_{j}$ coincides with its cell average over the cell $C_{j}$ and is also equal to the average of the values of $\widetilde{B}$ at the endpoints of $C_{j}$ , namely,

(5.7) $$\begin{eqnarray}B_{j}:=\widetilde{B}(x_{j})=\overline{\widetilde{B}}_{j}=\displaystyle \frac{1}{\unicode[STIX]{x1D6E5}x}\int _{C_{j}}\widetilde{B}(x)\,\text{d}x=\displaystyle \frac{B_{j+1/2}+B_{j-1/2}}{2}.\end{eqnarray}$$

Step 2: reconstruction of equilibrium variables. One of the key components of well-balanced schemes is performing a piecewise polynomial reconstruction of the equilibrium variables, $w$ and $q$ , rather than the conservative ones, $h$ and $q$ . To explain this, we notice that if at some time level $\overline{w}_{j}:=\,\overline{h}_{j}+B_{j}\equiv \widehat{w}$ and $\overline{q}_{j}\equiv 0$ for all $j$ , then all of the point values $w_{j+1/2}^{\pm }$ will assuredly have exactly the same value $\widehat{w}$ only if $w$ (which is constant at ‘lake at rest’ steady states) is reconstructed. If, instead of $w$ , the water depth $h$ is reconstructed, then it may happen that $h_{j+1/2}^{+}\neq h_{j+1/2}^{-}$ and hence $w_{j+1/2}^{+}=h_{j+1/2}^{+}+B_{j+1/2}\neq w_{j+1/2}^{-}=h_{j+1/2}^{-}+B_{j+1/2}$ , which will make the resulting reconstruction non-well-balanced.

Step 3: well-balanced discretization of the source term. After $w$ and $q$ are reconstructed, the point values of $q$ at the ‘lake at rest’ steady state (5.5) are $q_{j+1/2}^{\pm }\equiv 0$ and the point values of $h$ are obtained from $h_{j+1/2}^{\pm }=w_{j+1/2}^{\pm }-B_{j+1/2}=\widehat{w}-B_{j+1/2}$ . The fluxes at $x=x_{j+1/2}$ are then

$$\begin{eqnarray}\boldsymbol{F}(h_{j+1/2}^{\pm },q_{j+1/2}^{\pm })=\biggl(0,\displaystyle \frac{g}{2}(\widehat{w}-B_{j+1/2})^{2}\biggr)^{\top }.\end{eqnarray}$$

Thus, the numerical fluxes (3.15)–(3.17) at the ‘lake at rest’ steady state are

(5.8) $$\begin{eqnarray}{\mathcal{F}}_{j+1/2}^{\,(1)}=0\quad \text{and}\quad {\mathcal{F}}_{j+1/2}^{\,(2)}=\displaystyle \frac{g}{2}(\widehat{w}-B_{j+1/2})^{2},\end{eqnarray}$$

and the scheme (5.1) reduces to

(5.9) $$\begin{eqnarray}\displaystyle \displaystyle \frac{\text{d}}{\text{d}t}\,\overline{h}_{j} & = & \displaystyle 0,\quad \displaystyle \frac{\text{d}}{\text{d}t}\,\overline{q}_{j}=g(\widehat{w}-B_{j})\cdot \displaystyle \frac{B_{j+1/2}-B_{j-1/2}}{\unicode[STIX]{x1D6E5}x}+\,\overline{S}_{j}^{(2)},\end{eqnarray}$$

where

$$\begin{eqnarray}\overline{S}_{j}^{(2)}\approx -\displaystyle \frac{1}{\unicode[STIX]{x1D6E5}x}\int _{C_{j}}ghB_{x}\,\text{d}x.\end{eqnarray}$$

It is now clear that the right-hand side of (5.9) will vanish (and hence the designed central-upwind scheme will be well-balanced) if the following well-balanced quadrature is used to evaluate $\overline{S}_{j}^{(2)}$ :

(5.10) $$\begin{eqnarray}\overline{S}_{j}^{(2)}=-g\,\overline{h}_{j}\cdot \displaystyle \frac{B_{j+1/2}-B_{j-1/2}}{\unicode[STIX]{x1D6E5}x}.\end{eqnarray}$$

Note that this well-balanced quadrature can be, in fact, obtained by replacing $B_{x}(x_{j})$ in the midpoint quadrature (5.3) with its finite-difference approximation $(B_{j+1/2}-B_{j-1/2})/\unicode[STIX]{x1D6E5}x$ .

Remark 5.1. We would like to point out that the quadrature (5.10) is well-balanced not only for the central-upwind scheme, but for any semi-discrete scheme with the numerical flux satisfying (5.8) at ‘lake at rest’ steady states. This was first noticed in Audusse et al. (Reference Audusse, Bouchut, Bristeau, Klein and Perthame2004).

5.1.2 Moving-water equilibria-preserving central-upwind scheme

Designing moving-water equilibria-preserving central-upwind schemes is a more complicated task, which has recently been accomplished in Cheng and Kurganov (Reference Cheng and Kurganov2016) and Cheng et al. (Reference Cheng, Chertock, Herty, Kurganov and Wu2018) in two different ways. We now briefly describe the moving-water equilibria-preserving scheme from Cheng and Kurganov (Reference Cheng and Kurganov2016), which was designed following the key steps from Xing, Shu and Noelle (Reference Xing, Shu and Noelle2011).

Step 1: computation of $E_{j}$ . Given the cell averages $\overline{h}_{j}$ and $\overline{q}_{j}$ at a certain time level, we first compute

$$\begin{eqnarray}u_{j}=\displaystyle \frac{\overline{q}_{j}}{\overline{h}_{j}}\quad \text{and}\quad E_{j}=\displaystyle \frac{u_{j}^{2}}{2}+g(\,\overline{h}_{j}+B_{j}).\end{eqnarray}$$

Note that at the moving-water steady state (5.4), $E_{j}\equiv \widehat{E}$ , so that in this case, the equilibrium variables are $q$ and $E$ .

Step 2: reconstruction of equilibrium variables. We now reconstruct $q$ and $E$ instead of $q$ and $h$ (or $w$ ). This guarantees that if $E_{j}\equiv \widehat{E}$ and $\overline{q}_{j}\equiv 0$ for all $j$ , then all of the reconstructed point values satisfy $E_{j+1/2}^{\pm }\equiv \widehat{E}$ and $q_{j+1/2}^{\pm }\equiv 0$ .

Step 3: evaluation of point values of $h$ . After computing $q_{j+1/2}^{\pm }$ and $E_{j+1/2}^{\pm }$ , we recover the corresponding point values of $h$ by solving the following cubic equation:

(5.11) $$\begin{eqnarray}E_{j+1/2}^{\pm }=\displaystyle \frac{(q_{j+1/2}^{\pm })^{2}}{2(h_{j+1/2}^{\pm })^{2}}+g(h_{j+1/2}^{\pm }+B_{j+1/2})\end{eqnarray}$$

for $h_{j+1/2}^{\pm }$ . Details on how to find the unique physically relevant solution of (5.11) can be found in Cheng and Kurganov (Reference Cheng and Kurganov2016).

Step 4: well-balanced discretization of the source term. We now assume that the reconstructed point values satisfy $q_{j+1/2}^{\pm }\equiv \widehat{q}$ and $E_{j+1/2}^{\pm }\equiv \widehat{E}$ for all $j$ (note that this implies $h_{j+1/2}^{+}=h_{j+1/2}^{-}=:h_{j+1/2}$ for all $j$ ), and evaluate the corresponding central-upwind numerical fluxes (3.15)–(3.17):

(5.12) $$\begin{eqnarray}{\mathcal{F}}_{j+1/2}^{\,(1)}=\widehat{q}\quad \text{and}\quad {\mathcal{F}}_{j+1/2}^{\,(2)}=\displaystyle \frac{\widehat{q}^{\,2}}{h_{j+1/2}}+\displaystyle \frac{g}{2}h_{j+1/2}^{2}.\end{eqnarray}$$

We then take into account that $E_{j+1/2}=E_{j-1/2}$ , that is,

$$\begin{eqnarray}g(h_{j+1/2}-h_{j-1/2})=-\displaystyle \frac{\widehat{q}^{\,2}}{2}\biggl(\displaystyle \frac{1}{h_{j+1/2}^{2}}-\displaystyle \frac{1}{h_{j-1/2}^{2}}\biggr)-g(B_{j+1/2}-B_{j-1/2}),\end{eqnarray}$$

and substitute this together with (5.12) into the semi-discrete scheme (5.1) to obtain

(5.13a ) $$\begin{eqnarray}\displaystyle \displaystyle \frac{\text{d}}{\text{d}t}\,\overline{h}_{j} & = & \displaystyle 0,\end{eqnarray}$$

(5.13b ) $$\begin{eqnarray}\displaystyle \displaystyle \frac{\text{d}}{\text{d}t}\,\overline{q}_{j} & = & \displaystyle g\,\displaystyle \frac{h_{j+1/2}+h_{j-1/2}}{2}\cdot \displaystyle \frac{B_{j+1/2}-B_{j-1/2}}{\unicode[STIX]{x1D6E5}x}\nonumber\\ \displaystyle & & \displaystyle \qquad -\displaystyle \frac{h_{j+1/2}-h_{j-1/2}}{4\unicode[STIX]{x1D6E5}x}\biggl(\displaystyle \frac{\widehat{q}}{h_{j+1/2}}-\displaystyle \frac{\widehat{q}}{h_{j-1/2}}\biggr)^{2}+\,\overline{S}_{j}^{(2)}.\end{eqnarray}$$

Once again, we complete the construction of a well-balanced, moving-water equilibria-preserving central-upwind scheme by designing the well-balanced quadrature for $\overline{S}_{j}^{(2)}$ , which will guarantee that the right-hand side of (5.13) vanishes:

(5.14) $$\begin{eqnarray}\displaystyle \overline{S}_{j}^{(2)} & = & \displaystyle g\,\displaystyle \frac{h_{j+1/2}^{-}+h_{j-1/2}^{+}}{2}\cdot \displaystyle \frac{B_{j+1/2}-B_{j-1/2}}{\unicode[STIX]{x1D6E5}x}\nonumber\\ \displaystyle & & \displaystyle \qquad +\displaystyle \frac{h_{j+1/2}^{-}-h_{j-1/2}^{+}}{4\unicode[STIX]{x1D6E5}x}(u_{j+1/2}^{-}-u_{j-1/2}^{+})^{2}.\end{eqnarray}$$

Note that the obtained quadrature (5.14) is second-order accurate since the term $(u_{j+1/2}^{-}-u_{j-1/2}^{+})^{2}=O((\unicode[STIX]{x1D6E5}x)^{2})$ for smooth solutions.

5.1.3 Positivity-preserving central-upwind scheme

When the water surface $w$ is reconstructed (as in the still-water equilibria-preserving scheme described in Section 5.1.1), it may happen that at some cell $C_{j}$ either $w_{j+1/2}^{-}<B_{j+1/2}$ or $w_{j-1/2}^{+}<B_{j-1/2}$ even when $\overline{w}_{j}\geq B_{j}$ . Such a possibility is shown schematically in Figure 5.2, in which the first-order reconstruction $\widetilde{w}(x)$ is clearly below $\widetilde{B}(x)$ for some values of $x$ including several cell interfaces. Obviously, the use of conventional nonlinear limiters designed to minimize oscillations will not help to prevent the appearance of negative point values of $h$ in (almost) dry areas.

Figure 5.2.

Piecewise linear bottom topography approximant $\widetilde{B}$ and piecewise constant water surface reconstruction $\widetilde{w}$ . Notice that there are areas where $\widetilde{w}(x)<\widetilde{B}(x)$ and hence the water depth is negative.

In order to ensure that the reconstructed values of $h$ are non-negative, we take the following steps.

Step 1: positivity correction of the water surface reconstruction $\widetilde{w}$ . When a part of the linear piece of $\widetilde{w}$ in cell $C_{j}$ is below the corresponding linear piece of $\widetilde{B}$ , we need to modify the slope $(w_{x})_{j}$ to prevent the appearance of any negative values of $h$ . This can be done in several ways. In Kurganov and Petrova (Reference Kurganov and Petrova2007), we proposed the following simple correction algorithm (an alternative correction procedure will be described in Section 5.3 below):

(5.15a ) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{If }w_{j+1/2}^{-}<B_{j+1/2},\text{then take }(w_{x})_{j}=\displaystyle \frac{B_{j+1/2}-\,\overline{w}_{j}}{\unicode[STIX]{x1D6E5}x/2}\nonumber\\ \displaystyle & & \displaystyle \Longrightarrow \quad w_{j+1/2}^{-}=B_{j+1/2},\quad w_{j-1/2}^{+}=2\,\overline{w}_{j}-B_{j+1/2};\end{eqnarray}$$

(5.15b ) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{If }w_{j-1/2}^{+}<B_{j-1/2},\text{then take }(w_{x})_{j}=\displaystyle \frac{\overline{w}_{j}-B_{j-1/2}}{\unicode[STIX]{x1D6E5}x/2}\nonumber\\ \displaystyle & & \displaystyle \Longrightarrow \quad w_{j+1/2}^{-}=2\,\overline{w}_{j}-B_{j-1/2},\quad w_{j-1/2}^{+}=B_{j-1/2}.\end{eqnarray}$$

It is obvious that this correction procedure guarantees that the resulting reconstruction $\widetilde{w}$ will remain conservative and stay above the piecewise linear approximant of the bottom topography function $\widetilde{B}$ ; see Figure 5.3(b). Therefore, all of the corrected values of $h_{j+1/2}^{\pm }$ will be non-negative.

Figure 5.3.

Piece of the linear bottom topography approximant $\widetilde{B}$ together with either the originally reconstructed water surface (a) or positivity-preserving linear piece of $\widetilde{w}$ corrected using either (5.15) (b) or (5.34) (c).

Step 2: velocity desingularization. After the water surface correction performed in Step 1, all of the values of $h$ will be non-negative. However, they may be very small or even zero. This will not allow one to (accurately) compute the velocities $u_{j+1/2}^{\pm }$ , required in the computation of numerical flux and local speeds of propagation. In order to avoid the division by very small numbers, one should desingularize the velocity computation. This can be done in one of the following ways (for simplicity we omit the $\pm$ and $j\pm 1/2$ indexes):

(5.16) $$\begin{eqnarray}\displaystyle u & = & \displaystyle \left\{\begin{array}{@{}ll@{}}q/h\quad & \text{if }h\geq \unicode[STIX]{x1D700},\\ 0\quad & \text{otherwise},\end{array}\right.\end{eqnarray}$$

(5.17) $$\begin{eqnarray}\displaystyle u & = & \displaystyle \displaystyle \frac{2hq}{h^{2}+\max (h^{2},\unicode[STIX]{x1D700}^{2})},\end{eqnarray}$$

(5.18) $$\begin{eqnarray}\displaystyle u & = & \displaystyle \displaystyle \frac{\sqrt{2}\,hq}{\sqrt{h^{4}+\max (h^{4},\unicode[STIX]{x1D700}^{4})}},\end{eqnarray}$$

where $\unicode[STIX]{x1D700}$ is a small a priori chosen positive number (it should be selected based on some practical considerations that would suggest what values of $h$ may be considered negligibly small for each problem at hand). Each of the desingularization formulae (5.16)–(5.18) has its advantages and disadvantages; we refer the reader to Chertock et al. (Reference Chertock, Cui, Kurganov and Wu2015b ) and Kurganov and Petrova (Reference Kurganov and Petrova2007) for discussions on this matter.

As one can easily see, (5.16)–(5.18) reduce to $u=q/h$ for large values of $h$ , but when $h$ is small, the entire scheme will remain consistent only if we recompute the discharge using

$$\begin{eqnarray}q:=h\cdot u,\end{eqnarray}$$

where $u$ is computed by one of (5.16), (5.17) or (5.18).

Remark 5.2. Instead of desingularizing the computation of the velocity point values $u_{j+1/2}^{\pm }$ , one can implement an alternative approach: desingularize the computation of $u_{j}=\,\overline{q}_{j}/\,\overline{h}_{j}$ and then reconstruct a piecewise linear interpolant $\widetilde{u}$ . In this case, the point values of $u$ are obtained by

$$\begin{eqnarray}u_{j+1/2}^{+}=u_{j+1}-\displaystyle \frac{\unicode[STIX]{x1D6E5}x}{2}(u_{x})_{j+1},\quad u_{j+1/2}^{-}=u_{j}+\displaystyle \frac{\unicode[STIX]{x1D6E5}x}{2}(u_{x})_{j},\end{eqnarray}$$

where the slopes $(u_{x})_{j}$ are computed using a nonlinear limiter. The discharge point values are then computed using the reconstructed values of $h$ and $u$ , namely, by $q_{j+1/2}^{\pm }=h_{j+1/2}^{\pm }\cdot u_{j+1/2}^{\pm }$ .

We note that the resulting method will still be capable of exactly preserving ‘lake at rest’ steady states at which both $q\equiv 0$ and $u\equiv 0$ . For moving-water equilibria, however, $q\equiv \text{const.}$ , but $u\not \equiv \text{const.}$ , and thus one cannot reconstruct $u$ instead of $q$ while designing a moving-water equilibria-preserving schemes.

Step 3: adaptive time-step control. As was proved in Kurganov and Petrova (Reference Kurganov and Petrova2007, Theorem 2.1), if the resulting system of ODEs (5.1) is numerically solved using the forward Euler method, then all of the evolved cell averages of $h$ will be non-negative provided the CFL number is taken to be smaller than 1/2:

(5.19) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}t\leq \unicode[STIX]{x1D6E5}t_{\ast }:=\displaystyle \frac{\unicode[STIX]{x1D6E5}x}{2a},\quad a=\mathop{\max }_{j}\{a_{j+1/2}^{+},-a_{j-1/2}^{-}\},\end{eqnarray}$$

where $a_{j+1/2}^{\pm }$ are the local propagation speeds defined in (3.14). A similar result will be true for higher-order explicit SSP methods. However, as pointed out in Chertock et al. (Reference Chertock, Cui, Kurganov and Wu2015b ), the time steps should be selected adaptively in the following way (described here for the three-stage third-order SSP Runge–Kutta method).

While the solutions of (5.1) is evolved from time $t$ to $t+\unicode[STIX]{x1D6E5}t$ by the three-stage third-order SSP Runge–Kutta method, there will be two intermediate solutions, which we will denote by $\overline{\boldsymbol{U}}^{I}$ and $\overline{\boldsymbol{U}}^{\mathit{II}}$ , respectively. We would like to emphasize that Theorem 2.1 from Kurganov and Petrova (Reference Kurganov and Petrova2007) directly applies to the first stage of the three-stage third-order SSP Runge–Kutta method and hence the time-step restriction (5.19) guarantees the positivity of $\overline{h}_{j}^{I}$ for all $j$ provided $\overline{h}_{j}(t)\geq 0$ for all $j$ . The positivity of $\overline{h}_{j}^{\mathit{II}}$ and $\overline{h}_{j}(t+\unicode[STIX]{x1D6E5}t)$ will then be ensured by the same theorem provided $\unicode[STIX]{x1D6E5}t\leq \unicode[STIX]{x1D6E5}t_{\ast }^{I}$ and $\unicode[STIX]{x1D6E5}t\leq \unicode[STIX]{x1D6E5}t_{\ast }^{II}$ , where $\unicode[STIX]{x1D6E5}t_{\ast }^{I}$ and $\unicode[STIX]{x1D6E5}t_{\ast }^{II}$ are computed using (5.19) applied to the intermediate solutions $\overline{\boldsymbol{U}}^{I}$ and $\overline{\boldsymbol{U}}^{\mathit{II}}$ , respectively. In order to satisfy all of the aforementioned time-step restrictions, the following adaptive strategy should be implemented.

1. (i) Given the solution $\overline{\boldsymbol{U}}(t)$ , set $\unicode[STIX]{x1D6E5}t:=\unicode[STIX]{x1D705}\unicode[STIX]{x1D6E5}t_{\ast }$ , where $\unicode[STIX]{x1D705}\in (0,1)$ and $\unicode[STIX]{x1D6E5}t_{\ast }$ is given by (5.19).

2. (ii) Use $\unicode[STIX]{x1D6E5}t$ to compute $\overline{\boldsymbol{U}}^{I}$ at the first stage of the three-stage third-order SSP Runge–Kutta method.

3. (iii) Given the intermediate solution $\overline{\boldsymbol{U}}^{I}$ , compute $\unicode[STIX]{x1D6E5}t_{\ast }^{I}$ by (5.19).

4. (iv) If $\unicode[STIX]{x1D6E5}t_{\ast }^{I}<\unicode[STIX]{x1D6E5}t$ , set $\unicode[STIX]{x1D6E5}t:=\unicode[STIX]{x1D705}\unicode[STIX]{x1D6E5}t_{\ast }^{I}$ and go back to step (ii).

5. (v) Use $\unicode[STIX]{x1D6E5}t$ to compute $\overline{\boldsymbol{U}}^{\mathit{II}}$ at the second stage of the three-stage third-order SSP Runge–Kutta method.

6. (vi) Given the intermediate solution $\overline{\boldsymbol{U}}^{\mathit{II}}$ , compute $\unicode[STIX]{x1D6E5}t_{\ast }^{II}$ by (5.19).

7. (vii) If $\unicode[STIX]{x1D6E5}t_{\ast }^{II}<\unicode[STIX]{x1D6E5}t$ , set $\unicode[STIX]{x1D6E5}t:=\unicode[STIX]{x1D705}\unicode[STIX]{x1D6E5}t_{\ast }^{II}$ and go back to step (ii).

8. (viii) Use $\unicode[STIX]{x1D6E5}t$ to compute $\overline{\boldsymbol{U}}(t+\unicode[STIX]{x1D6E5}t)$ at the third stage of the three-stage third-order SSP Runge–Kutta method.

Note that in Chertock et al. (Reference Chertock, Cui, Kurganov and Wu2015b ), we have used $\unicode[STIX]{x1D705}=0.9$ .

5.1.4 Capturing wet/dry fronts

We would like to point out that when $h=0$ , the ‘lake at rest’ steady state (5.5) reduces to

(5.20) $$\begin{eqnarray}q\equiv 0,\quad h\equiv 0,\end{eqnarray}$$

which can be viewed as a ‘dry lake’. A good numerical scheme may be considered ‘truly’ well-balanced when it is capable of exactly preserving both ‘lake at rest’ and ‘dry lake’ steady states, as well as their combinations corresponding to the situations, in which the spatial domain $X$ is split into two non-overlapping parts $X_{1}$ (wet area) and $X_{2}$ (dry area) and the solution satisfies (5.5) in $X_{1}$ and (5.20) in $X_{2}$ .

Unfortunately, the central-upwind schemes presented above are not ‘truly’ well-balanced. The problem occurs at wet/dry fronts, where the water depth is very small and the reconstruction of $w$ is corrected. One way to design a ‘truly’ well-balanced central-upwind scheme is to modify the reconstruction of $w$ in partially flooded cells, as in Bollermann et al. (Reference Bollermann, Chen, Kurganov and Noelle2013). We now briefly describe this special well-balanced wet/dry reconstruction of the water surface.

Assuming at a certain time level $t$ , $\overline{w}_{j}\geq B_{j}$ for all $j$ , we define the following three types of computational cells.

• ∙ Fully flooded cell. If $\overline{w}_{j}\geq \max (\widehat{B}_{j-1/2},\widehat{B}_{j+1/2})$ and $\overline{h}_{j}:=\,\overline{w}_{j}-B_{j}>0$ , the cell is fully flooded.

• ∙ Partially flooded cell. If $B_{j}<\,\overline{w}_{j}<\max (\widehat{B}_{j-1/2},\widehat{B}_{j+1/2})$ , the cell is partially flooded.

• ∙ Dry cell. If $\overline{w}_{j}=B_{j}$ , the cell is dry.

The main idea of the well-balanced reconstruction in partially flooded cells is to allow sub-cell resolution there, that is, to allow the interpolant in these cells to consist of two linear pieces instead of just one as in fully flooded cells. For example, the first-order flat water surface reconstructions $w_{j}(x)$ in fully and partially flooded cells, schematically presented in Figure 5.4, can be written as

(5.21) $$\begin{eqnarray}w_{j}(x)=\left\{\begin{array}{@{}ll@{}}w_{j}\quad & \text{in the `wet' part of the cell},\\ \widetilde{B}(x)\quad & \text{in the `dry' part of the cell}.\end{array}\right.\end{eqnarray}$$

Here, $w_{j}=\,\overline{w}_{j}$ in fully flooded cells, and in partially flooded cells the value of $w_{j}$ is determined from the water conservation requirement stating that the area of the shaded triangle in Figure 5.4(b) is equal to $\unicode[STIX]{x1D6E5}x\cdot \overline{h}_{j}$ , which uniquely determines both $w_{j}$ and the location of the wet/dry interface $x_{w}^{\ast }$ ; see Bollermann et al. (Reference Bollermann, Chen, Kurganov and Noelle2013) for details.

Figure 5.4.

Well-balanced first-order water surface reconstruction in fully (a) and partially (b) flooded cells. $x_{w}^{\ast }$ is the reconstructed location of the wet/dry interface.

Before proceeding with the description of the second-order well-balanced wet/dry reconstruction of the water surface, we note that the positivity correction presented in Section 5.1.3 does not preserve ‘lake at rest’ steady states near wet/dry interfaces. In order to illustrate this, we show a typical situation in Figure 5.5. Let us assume that the actual water surface is flat so that in the fully flooded cell $C_{j+1}$ , $w_{j+1/2}^{+}=\,\overline{w}_{j}=\widehat{w}$ , but one can clearly see that the corrected value of $w_{j+1/2}^{-}$ is different from $\widehat{w}$ and hence some artificial waves will be generated at $x=x_{j+1/2}$ .

Figure 5.5.

Conservative and positivity-preserving, but not well-balanced piecewise linear reconstruction described in Section 5.1.3.

In order to make sure that the water surface interpolant in partially flooded cells respects the ‘lake at rest’ steady state, we proceed as follows. Let us assume that cell $C_{j}$ is partially flooded, cell $C_{j+1}$ is fully flooded, and that $B_{j-1/2}>\,\overline{w}_{j}>\widehat{B}_{j+1/2}$ (the case $B_{j+1/2}>\,\overline{w}_{j}>B_{j-1/2}$ can be treated similarly; in the case when cell $C_{j+1}$ is partially flooded, we simply use the first-order flat water surface reconstruction (5.21)), as shown in Figure 5.6. We then set the value of $w$ at $x=x_{j+1/2}$ to be $w_{j+1/2}^{-}:=w_{j+1/2}^{+}$ , and then use the water conservation requirement to uniquely determine one of the following.

1. (i) The water surface value $w_{j-1/2}^{+}$ in the case when the total amount of water is quite large (as in Figure 5.6(a)), which leads to the reconstruction in cell $C_{j}$ consisting of just one linear piece:

   $$\begin{eqnarray}\widetilde{w}(x)=w_{j-1/2}^{+}+(w_{j+1/2}^{+}-w_{j-1/2}^{+})\cdot \displaystyle \frac{x-x_{j-1/2}}{\unicode[STIX]{x1D6E5}x},\quad x\in C_{j}.\end{eqnarray}$$

2. (ii) The point $x_{R}^{\ast }\in C_{j}$ such that the total amount of water is quite small and can be placed in the interval $[x_{R}^{\ast },x_{j+1/2}]$ (as in Figure 5.6(b)), which leads to the reconstruction in cell $C_{j}$ consisting of the following two linear pieces:

   $$\begin{eqnarray}\widetilde{w}(x)=\left\{\begin{array}{@{}ll@{}}\widetilde{B}(x_{R}^{\ast })+(w_{j+1/2}^{+}-\widetilde{B}(x_{R}^{\ast }))\cdot {\textstyle \frac{x-x_{R}^{\ast }}{x_{j+1/2}-x_{R}^{\ast }}}\quad & \text{if}~x\in [x_{R}^{\ast },x_{j+1/2}],\\ \widetilde{B}(x)\quad & \text{if}~x\in [x_{j-1/2},x_{R}^{\ast }].\end{array}\right.\end{eqnarray}$$

We refer the reader to Bollermann et al. (Reference Bollermann, Chen, Kurganov and Noelle2013) for details.

Figure 5.6.

Well-balanced second-order water surface reconstruction in cases (i) (a) and (ii) (b). $x_{R}^{\ast }$ is the reconstructed location of the wet/dry interface. In both cases, the dashed line represents the corresponding well-balanced first-order water surface reconstructions.

5.1.5 Positivity-preserving via ‘draining’ time-step technique

While the original still-water equilibria-preserving central-upwind scheme from Kurganov and Petrova (Reference Kurganov and Petrova2007) described in Sections 5.1.1 and 5.1.3 is proved to be positivity-preserving, neither the moving-water equilibria-preserving central-upwind schemes from Cheng et al. (Reference Cheng, Chertock, Herty, Kurganov and Wu2018) and Cheng and Kurganov (Reference Cheng and Kurganov2016) (one of which is described in Section 5.1.2 above) nor the ‘truly’ well-balanced central-upwind scheme from Bollermann et al. (Reference Bollermann, Chen, Kurganov and Noelle2013) described in Section 5.1.4 are positivity-preserving. However, the positivity of the computed water depth can be enforced using the ‘draining’ time-step technique introduced in Bollermann et al. (Reference Bollermann, Noelle and Lukáčová-Medvid’ová2011) and briefly described below.

The key idea of the ‘draining’ time-step technique is to limit the outgoing fluxes in draining cells according to the following algorithm, which we present in the case of forward Euler time discretization, which for the first equation in (5.1) results in

(5.22) $$\begin{eqnarray}\overline{h}_{j}^{n+1}=\,\overline{h}_{j}^{n}-\displaystyle \frac{\unicode[STIX]{x1D6E5}t^{n}}{\unicode[STIX]{x1D6E5}x}({\mathcal{F}}_{j+1/2}^{(1)}-{\mathcal{F}}_{j-1/2}^{(1)}).\end{eqnarray}$$

Here, the numerical fluxes are evaluated at the time level $t=t^{n}$ and $\unicode[STIX]{x1D6E5}t^{n}$ is the time step restricted by (5.19).

In order to guarantee the positivity of $\overline{h}_{j}^{n+1}$ (assuming that $\overline{h}_{j}^{n}\geq 0$ for all $j$ , we first introduce the so-called ‘draining’ time step

$$\begin{eqnarray}\unicode[STIX]{x1D6E5}t_{j}^{\mathit{drain}}:=\displaystyle \frac{\,\overline{h}_{j}^{n}\unicode[STIX]{x1D6E5}x}{\max (0,{\mathcal{F}}_{j+1/2}^{(1)})+\max (0,-{\mathcal{F}}_{j-1/2}^{(1)})},\end{eqnarray}$$

which describes the time when the water contained in cell $C_{j}$ at the beginning of the time step would have left via the outflow fluxes. We then replace the evolution step (5.22) by

$$\begin{eqnarray}\overline{h}_{j}^{n+1}=\,\overline{h}_{j}^{n}-\displaystyle \frac{\unicode[STIX]{x1D6E5}t_{j+1/2}^{n}{\mathcal{F}}_{j+1/2}^{(1)}-\unicode[STIX]{x1D6E5}t_{j-1/2}^{n}{\mathcal{F}}_{j-1/2}^{(1)}}{\unicode[STIX]{x1D6E5}x},\end{eqnarray}$$

where we set the effective time steps at the cell interfaces to be

$$\begin{eqnarray}\unicode[STIX]{x1D6E5}t_{j+1/2}^{n}=\min (\unicode[STIX]{x1D6E5}t^{n},\unicode[STIX]{x1D6E5}t_{i}^{\mathit{drain}}),\quad i=j+\displaystyle \frac{1}{2}-\displaystyle \frac{\text{sgn}({\mathcal{F}}_{j+1/2}^{(1)})}{2}.\end{eqnarray}$$

This guarantees that $\overline{h}_{j}^{n+1}\geq 0$ for all $j$ .

Remark 5.3. We note that the ‘draining’ time-step technique can be applied to any Godunov-type finite-volume methods, not necessarily central-upwind schemes.

Remark 5.4. An extension of the ‘draining’ time-step technique to the 2-D case is quite straightforward.

5.2 Central-upwind schemes for the two-dimensional Saint-Venant system

In this section, we consider the 2-D Saint-Venant system (1.1), which is the 2-D system of balance laws (1.2) with

$$\begin{eqnarray}\boldsymbol{U}=\left(\begin{array}{@{}c@{}}h\\[2.0pt] q^{x}\\[2.0pt] q^{y}\end{array}\right),\quad \boldsymbol{F}(\boldsymbol{U})=\left(\begin{array}{@{}c@{}}q^{x}\\[2.0pt] {\textstyle \frac{(q^{x})^{2}}{h}}+{\textstyle \frac{gh^{2}}{2}}\\[2.0pt] {\textstyle \frac{q^{x}q^{y}}{h}}\end{array}\right),\quad \boldsymbol{G}(\boldsymbol{U})=\left(\begin{array}{@{}c@{}}q^{y}\\[2.0pt] {\textstyle \frac{q^{x}q^{y}}{h}}\\[2.0pt] {\textstyle \frac{(q^{y})^{2}}{h}}+{\textstyle \frac{gh^{2}}{2}}\end{array}\right),\end{eqnarray}$$

and $\boldsymbol{S}(h;B)=(0,-ghB_{x},-ghB_{y})^{\top }$ .

For the sake of simplicity, we will only consider central-upwind schemes on Cartesian grids as we have done in Section 4 above. We refer the reader to Bryson et al. (Reference Bryson, Epshteyn, Kurganov and Petrova2011) and Liu et al. (Reference Liu, Albright, Epshteyn and Kurganov2018), Shirkhani et al. (Reference Shirkhani, Mohammadian, Seidou and Kurganov2016) and Beljadid et al. (Reference Beljadid, Mohammadian and Kurganov2016) for the central-upwind schemes for the 2-D Saint-Venant system on triangular, quadrilateral and general polygonal grids, respectively.

A semi-discrete scheme for the system of balance laws (1.2) reads as

(5.23) $$\begin{eqnarray}\displaystyle \frac{\text{d}}{\text{d}t}\,\overline{\boldsymbol{U}}_{j,k}=-\displaystyle \frac{1}{\unicode[STIX]{x1D6E5}x}[\boldsymbol{{\mathcal{F}}}_{j+1/2,k}-\boldsymbol{{\mathcal{F}}}_{j-1/2,k}]-\displaystyle \frac{1}{\unicode[STIX]{x1D6E5}y}[\boldsymbol{{\mathcal{G}}}_{j,k+1/2}-\boldsymbol{{\mathcal{G}}}_{j,k-1/2}]+\,\overline{\boldsymbol{S}}_{j,k},\end{eqnarray}$$

where

(5.24) $$\begin{eqnarray}\overline{\boldsymbol{S}}_{j,k}\approx \displaystyle \frac{1}{\unicode[STIX]{x1D6E5}x\unicode[STIX]{x1D6E5}y}\iint _{C_{j,k}}\boldsymbol{S}(\boldsymbol{U})\,\text{d}x\,\text{d}y\end{eqnarray}$$

is an approximation of the source term cell average.

When the semi-discrete central-upwind scheme for the system of conservation laws (2.4) is extended to the system of balance laws (1.2), the fluxes are still given by (4.5)–(4.10), and one only needs to select an appropriate quadrature for the integral on the right-hand side of (5.24).

As in the 1-D case, a good well-balanced numerical scheme for the $\text{2-D}$ Saint-Venant system should be able to exactly preserve smooth steady-state solutions of (1.1), which generally have a much more complicated structure than their 1-D counterparts. However, the ‘lake at rest’ steady state can still be easily found explicitly and it is given by

(5.25) $$\begin{eqnarray}q^{x}\equiv q^{y}\equiv 0,\quad w=h+B\equiv \widehat{w}=\text{const.}\end{eqnarray}$$

In the rest of Section 5.2, we will refer to a central-upwind scheme as well-balanced if it is capable of exactly preserving (5.25).

5.2.1 Well-balanced central-upwind scheme

In order to design a well-balanced central-upwind scheme, we follow the main ideas presented in Kurganov and Levy (Reference Kurganov and Levy2002) and Kurganov and Petrova (Reference Kurganov and Petrova2007), and proceed as follows.

Step 1: piecewise bilinear reconstruction of the bottom topography. We start by replacing the original bottom topography function $B$ with its continuous piecewise bilinear approximation $\widetilde{B}$ , which at each cell $C_{j,k}$ is given by the bilinear form:

$$\begin{eqnarray}\displaystyle \widetilde{B}(x,y) & = & \displaystyle B_{j-1/2,k-1/2}+(B_{j+1/2,k-1/2}-B_{j-1/2,k-1/2})\cdot \displaystyle \frac{x-x_{j-1/2}}{\unicode[STIX]{x1D6E5}x}\nonumber\\ \displaystyle & & \displaystyle \qquad +(B_{j-1/2,k+1/2}-B_{j-1/2,k-1/2})\cdot \displaystyle \frac{y-y_{k-1/2}}{\unicode[STIX]{x1D6E5}y}\nonumber\\ \displaystyle & & \displaystyle \qquad +(B_{j+1/2,k+1/2}-B_{j+1/2,k-1/2}-B_{j-1/2,k+1/2}+B_{j-1/2,k-1/2})\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad \times \displaystyle \frac{(x-x_{j-1/2})(y-y_{k-1/2})}{\unicode[STIX]{x1D6E5}x\unicode[STIX]{x1D6E5}y},\quad (x,y)\in C_{j,k}.\nonumber\end{eqnarray}$$

Here, $B_{j\pm 1/2,k\pm 1/2}$ are the values of $\widetilde{B}$ at the corners of the cell $C_{j,k}$ , computed according to the following formula:

$$\begin{eqnarray}\displaystyle B_{j\pm 1/2,k\pm 1/2} & := & \displaystyle \displaystyle \frac{1}{2}\left(\,\max _{\unicode[STIX]{x1D709}^{2}+\unicode[STIX]{x1D702}^{2}=1}\lim _{h,\ell \rightarrow 0}B(x_{j\pm 1/2}+h\unicode[STIX]{x1D709},y_{k\pm 1/2}+\ell \unicode[STIX]{x1D702})\right.\nonumber\\ \displaystyle & & \displaystyle \left.\qquad +\min _{\unicode[STIX]{x1D709}^{2}+\unicode[STIX]{x1D702}^{2}=1}\lim _{h,\ell \rightarrow 0}B(x_{j\pm 1/2}+h\unicode[STIX]{x1D709},y_{k\pm 1/2}+\ell \unicode[STIX]{x1D702})\right),\nonumber\end{eqnarray}$$

which reduces to

$$\begin{eqnarray}B_{j\pm 1/2,k\pm 1/2}=B(x_{j\pm 1/2},y_{k\pm 1/2})\end{eqnarray}$$

if the function $B$ is continuous at $(x_{j\pm 1/2},y_{k\pm 1/2})$ .

Note that the restriction of the interpolant $\widetilde{B}$ along each of the lines $x=x_{j}$ or $y=y_{k}$ is a continuous piecewise linear function, and, as in the 1-D case (see (5.7)), the cell average of $\widetilde{B}$ over the cell $C_{j,k}$ is equal to its value at the centre of the cell and is also equal to the average of the values of $\widetilde{B}$ at the midpoints of the edges of $C_{j,k}$ . That is, we have

$$\begin{eqnarray}\displaystyle B_{j,k}:=\widetilde{B}(x_{j},y_{k}) & = & \displaystyle \overline{\widetilde{B}}_{j,k}=\displaystyle \frac{1}{\unicode[STIX]{x1D6E5}x\unicode[STIX]{x1D6E5}y}\iint _{C_{j,k}}\widetilde{B}(x,y)\,\text{d}x\,\text{d}y\nonumber\\ \displaystyle & = & \displaystyle \displaystyle \frac{1}{4}(B_{j+1/2,k}+B_{j-1/2,k}+B_{j,k+1/2}+B_{j,k-1/2}),\nonumber\end{eqnarray}$$

where

$$\begin{eqnarray}B_{j+1/2,k}:=\widetilde{B}(x_{j+1/2},y_{k})=\displaystyle \frac{1}{2}(B_{j+1/2,k+1/2}+B_{j+1/2,k-1/2})\end{eqnarray}$$

and

$$\begin{eqnarray}B_{j,k+1/2}:=\widetilde{B}(x_{j},y_{k+1/2})=\displaystyle \frac{1}{2}(B_{j+1/2,k+1/2}+B_{j-1/2,k+1/2}).\end{eqnarray}$$

Remark 5.5. We would like to point out that when a triangular grid is used (as in Bryson et al. Reference Bryson, Epshteyn, Kurganov and Petrova2011, Liu et al. Reference Liu, Albright, Epshteyn and Kurganov2018), the bottom topography function should be approximated using the continuous piecewise linear interpolant uniquely determined by the point values of $B$ at the triangular cell vertices. In the case of general quadrilateral (Shirkhani et al. Reference Shirkhani, Mohammadian, Seidou and Kurganov2016) or polygonal (Beljadid et al. Reference Beljadid, Mohammadian and Kurganov2016) grids, the finite-volume cell may be split into several triangular sub-cells and then the bottom topography function can be approximated using a continuous piecewise linear interpolant as well.

Step 2: reconstruction of equilibrium variables. As in the 1-D case, we reconstruct the equilibrium variables, $w$ , $q^{x}$ and $q^{y}$ , rather than the conservative ones, $h$ , $q^{x}$ and $q^{y}$ .

Step 3: well-balanced discretization of the source term. After $w$ , $q^{x}$ and $q^{y}$ are reconstructed, the point values of $q^{x}$ and $q^{y}$ at the ‘lake at rest’ steady state (5.25) are

$$\begin{eqnarray}\displaystyle (q_{j,k}^{x})^{\text{E}}\equiv (q_{j,k}^{x})^{\text{E}}\equiv (q_{j,k}^{x})^{\text{N}}\equiv (q_{j,k}^{x})^{\text{S}} & \equiv & \displaystyle 0,\nonumber\\ \displaystyle (q_{j,k}^{y})^{\text{E}}\equiv (q_{j,k}^{y})^{\text{E}}\equiv (q_{j,k}^{y})^{\text{N}}\equiv (q_{j,k}^{y})^{\text{S}} & \equiv & \displaystyle 0,\nonumber\end{eqnarray}$$

and the point values of $h$ are given by

$$\begin{eqnarray}\displaystyle h_{j,k}^{\text{E}} & = & \displaystyle w_{j,k}^{\text{E}}-B_{j+1/2,k}=\widehat{w}-B_{j+1/2,k},\nonumber\\ \displaystyle h_{j,k}^{\text{W}} & = & \displaystyle w_{j,k}^{\text{W}}-B_{j-1/2,k}=\widehat{w}-B_{j-1/2,k},\nonumber\\ \displaystyle h_{j,k}^{\text{N}} & = & \displaystyle w_{j,k}^{\text{N}}-B_{j,k+1/2}=\widehat{w}-B_{j,k+1/2},\nonumber\\ \displaystyle h_{j,k}^{\text{S}} & = & \displaystyle w_{j,k}^{\text{S}}-B_{j,k-1/2}=\widehat{w}-B_{j,k-1/2}.\nonumber\end{eqnarray}$$

The fluxes at $(x_{j+1/2},y_{k})$ and $(x_{j},y_{k+1/2})$ are then

$$\begin{eqnarray}\displaystyle \boldsymbol{F}(\boldsymbol{U}_{j,k}^{\text{E}}) & = & \displaystyle \boldsymbol{F}(\boldsymbol{U}_{j+1,k}^{\text{W}})=\biggl(0,\displaystyle \frac{g}{2}(\widehat{w}-B_{j+1/2,k})^{2},0\biggr)^{\top },\nonumber\\ \displaystyle \boldsymbol{G}(\boldsymbol{U}_{j,k}^{\text{N}}) & = & \displaystyle \boldsymbol{G}(\boldsymbol{U}_{j,k+1}^{\text{S}})=\biggl(0,0,\displaystyle \frac{g}{2}(\widehat{w}-B_{j,k+1/2})^{2}\biggr)^{\top }.\nonumber\end{eqnarray}$$

Thus the numerical fluxes (4.5)–(4.10) at the ‘lake at rest’ steady state are

$$\begin{eqnarray}\displaystyle \begin{array}{@{}rclrc@{}}{\mathcal{F}}_{j+1/2,k}^{\,(1)} & = & {\mathcal{F}}_{j+1/2,k}^{\,(3)}=0,\quad {\mathcal{F}}_{j+1/2,k}^{\,(2)} & = & \displaystyle \frac{g}{2}(\widehat{w}-B_{j+1/2,k})^{2},\\ {\mathcal{G}}_{j+1/2,k}^{\,(1)} & = & {\mathcal{G}}_{j+1/2,k}^{\,(2)}=0,\quad {\mathcal{G}}_{j+1/2,k}^{\,(3)} & = & \displaystyle \frac{g}{2}(\widehat{w}-B_{j,k+1/2})^{2},\end{array} & & \displaystyle \nonumber\end{eqnarray}$$

and the scheme (5.23) reduces to

(5.26a ) $$\begin{eqnarray}\displaystyle \displaystyle \frac{\text{d}}{\text{d}t}\,\overline{h}_{j,k} & = & \displaystyle 0,\end{eqnarray}$$

(5.26b ) $$\begin{eqnarray}\displaystyle \displaystyle \frac{\text{d}}{\text{d}t}\,\overline{q}_{j,k}^{\,x} & = & \displaystyle g(\widehat{w}-B_{j,k})\cdot \displaystyle \frac{B_{j+1/2,k}-B_{j-1/2,k}}{\unicode[STIX]{x1D6E5}x}+\,\overline{S}_{j,k}^{(2)},\end{eqnarray}$$

(5.26c ) $$\begin{eqnarray}\displaystyle \displaystyle \frac{\text{d}}{\text{d}t}\,\overline{q}_{j,k}^{\,y} & = & \displaystyle g(\widehat{w}-B_{j,k})\cdot \displaystyle \frac{B_{j,k+1/2}-B_{j,k-1/2}}{\unicode[STIX]{x1D6E5}y}+\,\overline{S}_{j,k}^{(3)},\end{eqnarray}$$

where

$$\begin{eqnarray}\overline{S}_{j,k}^{(2)}\approx -\displaystyle \frac{1}{\unicode[STIX]{x1D6E5}x\unicode[STIX]{x1D6E5}y}\int _{C_{j,k}}ghB_{x}\,\text{d}x\,\text{d}y,\quad \overline{S}_{j,k}^{(3)}\approx -\displaystyle \frac{1}{\unicode[STIX]{x1D6E5}x\unicode[STIX]{x1D6E5}y}\int _{C_{j,k}}ghB_{y}\,\text{d}x\,\text{d}y.\end{eqnarray}$$

As in the 1-D case, the right-hand side of (5.26) must vanish to ensure that the designed central-upwind scheme is well-balanced, and therefore the following well-balanced quadratures are used to evaluate $\overline{S}_{j,k}^{(2)}$ and $\overline{S}_{j,k}^{(3)}$ :

$$\begin{eqnarray}\overline{S}_{j,k}^{(2)}=-g\,\overline{h}_{j,k}\cdot \displaystyle \frac{B_{j+1/2,k}-B_{j-1/2,k}}{\unicode[STIX]{x1D6E5}x},\quad \overline{S}_{j,k}^{(3)}=-g\,\overline{h}_{j,k}\cdot \displaystyle \frac{B_{j,k+1/2}-B_{j,k-1/2}}{\unicode[STIX]{x1D6E5}y}.\end{eqnarray}$$

Remark 5.6. We stress that development of a well-balanced quadrature for the source term discretization on triangular grids is a more delicate task. One first needs to apply Green’s formula to the vector field $({\textstyle \frac{1}{2}}(w-B)^{2},0)^{\top }$ , and only then does it become clear how to identify the desired quadrature; see Bryson et al. (Reference Bryson, Epshteyn, Kurganov and Petrova2011) and Liu et al. (Reference Liu, Albright, Epshteyn and Kurganov2018) for details.

5.2.2 Positivity-preserving central-upwind scheme

As in the 1-D case, the original water surface reconstruction may produce negative values of $h=w-B$ . In order to ensure that the reconstructed values of $h$ are non-negative, we take the same three steps as in Section 5.1.3.

First, we correct the reconstruction of $w$ by modifying the slopes $(w_{x})_{j,k}$ and $(w_{y})_{j,k}$ as follows:

$$\begin{eqnarray}\displaystyle & & \displaystyle \text{If }w_{j,k}^{\text{E}}<B_{j+1/2,k},\text{then take }(w_{x})_{j,k}=\displaystyle \frac{B_{j+1/2,k}-\,\overline{w}_{j,k}}{\unicode[STIX]{x1D6E5}x/2}\nonumber\\ \displaystyle & & \displaystyle \Longrightarrow \quad w_{j,k}^{\text{E}}=B_{j+1/2,k},\quad w_{j,k}^{\text{W}}=2\,\overline{w}_{j,k}-B_{j+1/2,k};\nonumber\\ \displaystyle & & \displaystyle \text{If }w_{j,k}^{\text{W}}<B_{j-1/2,k},\text{then take }(w_{x})_{j,k}=\displaystyle \frac{\overline{w}_{j,k}-B_{j-1/2,k}}{\unicode[STIX]{x1D6E5}x/2}\nonumber\\ \displaystyle & & \displaystyle \Longrightarrow \quad w_{j,k}^{\text{E}}=2\,\overline{w}_{j,k}-B_{j-1/2,k},\quad w_{j,k}^{\text{W}}=B_{j-1/2,k};\nonumber\\ \displaystyle & & \displaystyle \text{If }w_{j,k}^{\text{N}}<B_{j,k+1/2},\text{then take }(w_{y})_{j,k}=\displaystyle \frac{B_{j,k+1/2}-\,\overline{w}_{j,k}}{\unicode[STIX]{x1D6E5}y/2}\nonumber\\ \displaystyle & & \displaystyle \Longrightarrow \quad w_{j,k}^{\text{N}}=B_{j,k+1/2},\quad w_{j,k}^{\text{S}}=2\,\overline{w}_{j,k}-B_{j,k+1/2};\nonumber\\ \displaystyle & & \displaystyle \text{If }w_{j,k}^{\text{S}}<B_{j,k-1/2},\text{then take }(w_{y})_{j,k}=\displaystyle \frac{\overline{w}_{j,k}-B_{j,k-1/2}}{\unicode[STIX]{x1D6E5}y/2}\nonumber\\ \displaystyle & & \displaystyle \Longrightarrow \quad w_{j,k}^{\text{N}}=2\,\overline{w}_{j,k}-B_{j,k-1/2},\quad w_{j,k}^{\text{S}}=B_{j,k-1/2}.\nonumber\end{eqnarray}$$

This correction procedure guarantees that the resulting reconstruction $\widetilde{w}$ will remain conservative and its restrictions on the lines $y=y_{k}$ and $x=x_{j}$ will be above $\widetilde{B}(x,y_{k})$ and $\widetilde{B}(x_{j},y)$ , respectively. Hence, all of the corrected values $h_{j,k}^{\text{E}}=w_{j,k}^{\text{E}}-B_{j+1/2,k}$ , $h_{j,k}^{\text{W}}=w_{j,k}^{\text{W}}-B_{j-1/2,k}$ , $h_{j,k}^{\text{N}}=w_{j,k}^{\text{N}}-B_{j,k+1/2}$ and $h_{j,k}^{\text{S}}=w_{j,k}^{\text{S}}-B_{j,k-1/2}$ will be non-negative. Notice that unlike the 1-D case, this does not guarantee the non-negativity of $\widetilde{w}-\widetilde{B}$ in the entire cell $C_{j,k}$ . However, this is not a problem since our goal is to preserve positivity of the cell averages of $h$ and its point values used in the scheme ( $h_{j,k}^{\text{E}}$ , $h_{j,k}^{\text{W}}$ , $h_{j,k}^{\text{S}}$ and $h_{j,k}^{\text{N}}$ ).

Remark 5.7. In the case of a triangular grid, the positivity correction is performed in a different way: one should make sure that the point values of $w$ at the cell vertices are above the values of $\widetilde{B}$ there, and this guarantees that the entire corrected linear piece of $w$ will stay above the corresponding linear piece of $\widetilde{B}$ . We refer the reader to Bryson et al. (Reference Bryson, Epshteyn, Kurganov and Petrova2011) and Liu et al. (Reference Liu, Albright, Epshteyn and Kurganov2018) for details.

Second, we desingularize the velocity computation using $u=q^{x}/h$ and $v=q^{y}/h$ (this is done exactly as in the 1-D case described in Section 5.1.3). Third, we implement the same adaptive time-step control described in Section 5.1.3. However, the basic time-step bound that guarantees non-negativity of $h$ is more restrictive (one has to use the CFL number to be less than or equal to 1/4 instead of 1/2 used in the 1-D case), and one has to choose

$$\begin{eqnarray}\unicode[STIX]{x1D6E5}t\leq \unicode[STIX]{x1D6E5}t_{\ast }:=\min \biggl\{\displaystyle \frac{\unicode[STIX]{x1D6E5}x}{4a},\displaystyle \frac{\unicode[STIX]{x1D6E5}y}{4b}\biggr\},\end{eqnarray}$$

where

$$\begin{eqnarray}a=\max _{j,k}\{a_{j+1/2,k}^{+},-a_{j+1/2,k}^{-}\},\quad b=\max _{j,k}\{b_{j,k+1/2}^{+},-b_{j,k+1/2}^{-}\},\end{eqnarray}$$

and $a_{j+1/2,k}^{\pm }$ and $b_{j,k+1/2}^{\pm }$ are the local propagation speeds defined in (4.3).

5.3 Central-upwind schemes for the Saint-Venant systems with friction terms

In this section, we consider a more general shallow-water system, which is obtained by taking into account the bottom friction terms. In the 2-D case, the studied system reads as

(5.27a ) $$\begin{eqnarray}\displaystyle h_{t}+(hu)_{x}+(hv)_{y} & = & \displaystyle 0,\end{eqnarray}$$

(5.27b ) $$\begin{eqnarray}\displaystyle (hu)_{t}+\biggl(hu^{2}+\displaystyle \frac{g}{2}h^{2}\biggr)_{x}+(huv)_{y} & = & \displaystyle -ghB_{x}-ghI^{x},\end{eqnarray}$$

(5.27c ) $$\begin{eqnarray}\displaystyle (hv)_{t}+(huv)_{x}+\biggl(hv^{2}+\displaystyle \frac{g}{2}h^{2}\biggr)_{y} & = & \displaystyle -ghB_{y}-ghI^{y},\end{eqnarray}$$

$I^{x}$

$I^{y}$

(5.28) $$\begin{eqnarray}I^{x}=\displaystyle \frac{n^{2}}{h^{4/3}}\,u\sqrt{u^{2}+v^{2}},\quad I^{y}=\displaystyle \frac{n^{2}}{h^{4/3}}\,v\sqrt{u^{2}+v^{2}},\end{eqnarray}$$

where $n$ is the Manning coefficient. Notice that if $h\approx 0$ , the friction terms (5.28) become stiff damping terms, and this increases the level of complexity in the development of efficient numerical methods for the system (5.27), (5.28). Another difficulty is related to the fact that this system admits not only ‘lake at rest’ steady states (5.25), but other steady-state solutions, which may become more relevant in many practical situations, for example, when drainage of rain water in urban areas is simulated; see, for example, Cea, Garrido and Puertas (Reference Cea, Garrido and Puertas2010) and Cea and Vázquez-Cendón (Reference Cea and Vázquez-Cendón2012) and references therein.

Let us first consider the simplest 1-D case, in which the system (5.27), (5.28) reduces to

(5.29a ) $$\begin{eqnarray}\displaystyle h_{t}+(hu)_{x} & = & \displaystyle 0,\end{eqnarray}$$

(5.29b ) $$\begin{eqnarray}\displaystyle (hu)_{t}+\biggl(hu^{2}+\displaystyle \frac{g}{2}h^{2}\biggr)_{x} & = & \displaystyle -ghB_{x}-g\displaystyle \frac{n^{2}}{h^{1/3}}|u|u.\end{eqnarray}$$

$u\neq 0$

(5.30) $$\begin{eqnarray}hu\equiv \text{const.},\quad h\equiv \text{const.},\quad B_{x}\equiv \text{const.}\end{eqnarray}$$

This solution corresponds to the situation when the water flows over a slanted infinitely long surface with a constant slope. The structure of $\text{2-D}$ steady states is substantially more complicated. However, the quasi-1-D steady-state solutions

(5.31) $$\begin{eqnarray}h\equiv \text{const.},\quad hu\equiv \text{const.},\quad hv\equiv 0,\quad B_{x}\equiv \text{const.},\quad B_{y}\equiv 0\end{eqnarray}$$

and

(5.32) $$\begin{eqnarray}h\equiv \text{const.},\quad hu\equiv 0,\quad hv\equiv \text{const.},\quad B_{x}\equiv 0,\quad B_{y}\equiv \text{const.}\end{eqnarray}$$

are still physically relevant.

A well-balanced Roe-type numerical scheme which is capable of exactly preserving the steady states (5.30), (5.31) and (5.32) was proposed in Cea and Vázquez-Cendón (Reference Cea and Vázquez-Cendón2012). However, to maintain the positivity of the water depth $h$ , the scheme in Cea and Vázquez-Cendón (Reference Cea and Vázquez-Cendón2012) may require one to use very small time steps and thus may not be robust in certain settings. Another Godunov-type scheme for the 1-D system (5.29) was proposed in Berthon, Marche and Turpault (Reference Berthon, Marche and Turpault2011). Although this method does not suffer from restrictive time-stepping, it is capable of preserving ‘lake at rest’ steady states only.

In Chertock et al. (Reference Chertock, Cui, Kurganov and Wu2015b ), we have developed well-balanced positivity-preserving central-upwind schemes for both 1-D (5.29) and 2-D (5.27) systems. For the sake of brevity, we will now describe the 1-D scheme only.

The 1-D semi-discrete central-upwind scheme for (5.29) is still given by (5.1), but the discrete source term now consists of approximations of the following two integrals:

(5.33) $$\begin{eqnarray}\overline{S}_{j}^{(2)}\approx -\displaystyle \frac{1}{\unicode[STIX]{x1D6E5}x}\int _{C_{j}}ghB_{x}\,\text{d}x-\displaystyle \frac{1}{\unicode[STIX]{x1D6E5}x}\int _{C_{j}}\displaystyle \frac{gn^{2}}{h^{1/3}}|u|u\,\text{d}x.\end{eqnarray}$$

One can easily show that the well-balanced quadrature for the first integral on the right-hand side of (5.33) is still given by (5.10) and the well-balanced quadrature for the second integral on the right-hand side of (5.33) is obtained using the midpoint rule and one of the desingularization formulae (5.16), (5.17) or (5.18). In Chertock et al. (Reference Chertock, Cui, Kurganov and Wu2015b ), the desingularization formula (5.17) has been used and the resulting quadrature is

$$\begin{eqnarray}\displaystyle \frac{1}{\unicode[STIX]{x1D6E5}x}\int _{C_{j}}\displaystyle \frac{gn^{2}}{h^{1/3}}|u|u\,\text{d}x\approx g\,n^{2}\biggl(\displaystyle \frac{2\,\overline{h}_{j}}{\overline{h}_{j}^{2}+\max (\,\overline{h}_{j}^{2},\unicode[STIX]{x1D700}^{2})}\biggr)^{7/3}|\,\overline{q}_{j}|\,\overline{q}_{j}.\end{eqnarray}$$

In order to complete the construction of the semi-discrete central-upwind scheme, we proceed along the lines of Sections 3.3 and 5.1. The resulting scheme, however, will have two major drawbacks. First, it will be capable of preserving ‘lake at rest’ steady states only. Second, it will be very inefficient in the case when some (almost) dry areas are present, as the friction term may become very stiff there and explicit SSP Runge–Kutta methods will have very severe time-step stability restriction. We improve the central-upwind scheme by taking the following two steps.

Step 1: modified positivity correction of the water surface reconstruction $\widetilde{w}$ . As one can easily see, the water surface positivity correction procedure described in Section 5.1.3 does not preserve the water surface profile that correspond to the steady state (5.30); see Figure 5.3(b). Therefore, in Chertock et al. (Reference Chertock, Cui, Kurganov and Wu2015b ) we have proposed an alternative positivity correction procedure presented in Figure 5.3(c) and described here.

In the cells, where the original limited water surface reconstruction produces negative values of $h$ , we make the slopes of $w$ equal to the corresponding slopes of $B$ . Namely, we proceed as follows:

(5.34) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{If }w_{j+1/2}^{-}<B_{j+1/2}\text{ or }w_{j-1/2}^{+}<B_{j-1/2},\text{then take }(w_{x})_{j}=(B_{x})_{j}\nonumber\\ \displaystyle & & \displaystyle \Longrightarrow \quad w_{j+1/2}^{-}=\,\overline{h}_{j}+B_{j+1/2},\quad w_{j-1/2}^{+}=\overline{h}_{j}+B_{j-1/2}.\end{eqnarray}$$

This correction (unlike the correction procedure described in Sections 5.1.3 and its more sophisticated modification described in Sections 5.1.4) will not only guarantee the positivity of $h_{j+1/2}^{\pm }$ but will also be able to exactly reconstruct the steady-state solution (5.30).

Step 2: steady state and sign preserving semi-implicit Runge–Kutta methods. As mentioned earlier, in the presence of dry and/or almost dry areas, the explicit treatment of the friction terms imposes a severe time-step restriction, which may be several orders of magnitude smaller than a typical time step used for the corresponding friction-free Saint-Venant system.

An attractive alternative to explicit methods is that of implicit–explicit (IMEX) SSP Runge–Kutta solvers, which treat the stiff part of the underlying ODE system implicitly and thus typically have the stability domains based on the non-stiff term only; see, for example, Ascher, Ruuth and Spiteri (Reference Ascher, Ruuth and Spiteri1997), Higueras and Roldán (Reference Higueras and Roldán2006), Hundsdorfer and Ruuth (Reference Hundsdorfer and Ruuth2007), Higueras, Happenhofer, Koch and Kupka (Reference Higueras, Happenhofer, Koch and Kupka2014) and Pareschi and Russo (Reference Pareschi and Russo2001, Reference Pareschi and Russo2005). However, a straightforward implementation of these methods may break the discrete balance between the fluxes, geometric source and the friction terms maintained by the derived semi-discrete central-upwind scheme, and the resulting fully discrete method will not be able to preserve the relevant steady states and the positivity of the computed water depth.

In order to overcome this difficulty, we have recently developed a family of second-order semi-implicit time integration methods for systems of ODEs with stiff damping term; see Chertock, Cui, Kurganov and Tong (Reference Chertock, Cui, Kurganov and Tong2015a ). In these methods, only a portion of the stiff term is implicitly treated, and therefore the evolution equation is very easy to solve and implement compared to fully implicit or IMEX methods. The important feature of the ODE solvers introduced in Chertock et al. (Reference Chertock, Cui, Kurganov and Tong2015a ) resides in the fact that they are capable of exactly preserving the steady states as well as maintaining the sign of the computed solution under the time step restriction determined by the non-stiff part of the system only. These semi-implicit methods are based on the modification of explicit SSP Runge–Kutta methods and are proven to have a formal second order of accuracy, $A(\unicode[STIX]{x1D6FC})$ -stability and stiff decay. We now briefly describe the application of the second-order semi-implicit ODE solver SI-RK3 from Chertock et al. (Reference Chertock, Cui, Kurganov and Tong2015a ) to the ODE system (5.1), (3.15)–(3.17), (5.33).

We first introduce the grid function of the numerical solution $\overline{\boldsymbol{U}}:=\{\,\overline{\boldsymbol{U}}_{j}\}$ . We then denote the discretization of the sum of fluxes and geometric source term on the right-hand side of (5.1) by

$$\begin{eqnarray}\displaystyle {\mathcal{L}}^{(1)}[\,\overline{\boldsymbol{U}}]_{j} & := & \displaystyle -\displaystyle \frac{{\mathcal{F}}_{j+1/2}^{(1)}-{\mathcal{F}}_{j-1/2}^{(1)}}{\unicode[STIX]{x1D6E5}x},\nonumber\\ \displaystyle {\mathcal{L}}^{(2)}[\,\overline{\boldsymbol{U}}]_{j} & := & \displaystyle -\displaystyle \frac{{\mathcal{F}}_{j+1/2}^{(2)}-{\mathcal{F}}_{j-1/2}^{(2)}}{\unicode[STIX]{x1D6E5}x}-g\,\overline{h}_{j}\displaystyle \frac{B_{j+1/2}-B_{j-1/2}}{\unicode[STIX]{x1D6E5}x},\nonumber\end{eqnarray}$$

and introduce the discrete friction coefficient

$$\begin{eqnarray}{\mathcal{M}}(\,\overline{\boldsymbol{U}}_{j}):=-gn^{2}\biggl(\displaystyle \frac{2\,\overline{h}_{j}}{\,\overline{h}_{j}^{2}+\max (\,\overline{h}_{j}^{2},\unicode[STIX]{x1D700}^{2})}\biggr)^{7/3}|\,\overline{q}_{j}|,\end{eqnarray}$$

so that the ODE system (5.1) can be rewritten as

(5.35) $$\begin{eqnarray}\displaystyle \frac{\text{d}}{\text{d}t}\,\overline{h}_{j}={\mathcal{L}}^{(1)}[\,\overline{\boldsymbol{U}}]_{j},\quad \displaystyle \frac{\text{d}}{\text{d}t}\,\overline{q}_{j}={\mathcal{L}}^{(2)}[\,\overline{\boldsymbol{U}}]_{j}+{\mathcal{M}}(\,\overline{\boldsymbol{U}}_{j})\,\overline{q}_{j}.\end{eqnarray}$$

We now implement the SI-RK3 method to the system (5.35). (The SI-RK3 method is a second-order semi-implicit Runge–Kutta method based on the three-stage third-order SSP Runge–Kutta method; for details, see Chertock et al. (Reference Chertock, Cui, Kurganov and Tong2015a , Section 3).) The resulting fully discrete scheme can be written as

(5.36a ) $$\begin{eqnarray}\displaystyle \overline{h}_{j}^{I} & = & \displaystyle \,\overline{h}_{j}(t)+\unicode[STIX]{x1D6E5}t{\mathcal{L}}^{(1)}[\,\overline{\boldsymbol{U}}(t)]_{j},\end{eqnarray}$$

(5.36b ) $$\begin{eqnarray}\displaystyle \,\overline{q}_{j}^{I} & = & \displaystyle \displaystyle \frac{\,\overline{q}_{j}(t)+\unicode[STIX]{x1D6E5}t{\mathcal{L}}^{(2)}[\,\overline{\boldsymbol{U}}(t)]_{j}}{1-\unicode[STIX]{x1D6E5}t\,{\mathcal{M}}(\,\overline{\boldsymbol{U}}_{j}(t))},\end{eqnarray}$$

(5.36c ) $$\begin{eqnarray}\displaystyle \overline{h}_{j}^{\mathit{II}} & = & \displaystyle \displaystyle \frac{3}{4}\,\overline{h}_{j}(t)+\displaystyle \frac{1}{4}(\,\overline{h}_{j}^{I}+\unicode[STIX]{x1D6E5}t{\mathcal{L}}^{(1)}[\,\overline{\boldsymbol{U}}^{I}]_{j}),\end{eqnarray}$$

(5.36d ) $$\begin{eqnarray}\displaystyle \overline{q}_{j}^{\,\mathit{II}} & = & \displaystyle \displaystyle \frac{3}{4}\,\overline{q}_{j}(t)+\displaystyle \frac{1}{4}\cdot \displaystyle \frac{\overline{q}_{j}^{\,I}+\unicode[STIX]{x1D6E5}t{\mathcal{L}}^{(2)}[\,\overline{\boldsymbol{U}}^{I}]_{j}}{1-\unicode[STIX]{x1D6E5}t\,{\mathcal{M}}(\,\overline{\boldsymbol{U}}_{j}^{I})},\end{eqnarray}$$

(5.36e ) $$\begin{eqnarray}\displaystyle \overline{h}_{j}^{\mathit{III}} & = & \displaystyle \displaystyle \frac{1}{3}\,\overline{h}_{j}(t)+\displaystyle \frac{2}{3}(\,\overline{h}_{j}^{\mathit{II}}+\unicode[STIX]{x1D6E5}t{\mathcal{L}}^{(1)}[\,\overline{\boldsymbol{U}}^{\mathit{II}}]_{j}),\end{eqnarray}$$

(5.36f ) $$\begin{eqnarray}\displaystyle \overline{q}_{j}^{\,\mathit{III}} & = & \displaystyle \displaystyle \frac{1}{3}\,\overline{q}_{j}(t)+\displaystyle \frac{2}{3}\cdot \displaystyle \frac{\,\overline{q}_{j}^{\,\mathit{II}}+\unicode[STIX]{x1D6E5}t{\mathcal{L}}^{(2)}[\,\overline{\boldsymbol{U}}^{\mathit{II}}]_{j}}{1-\unicode[STIX]{x1D6E5}t\,{\mathcal{M}}(\,\overline{\boldsymbol{U}}_{j}^{\mathit{II}})},\end{eqnarray}$$

(5.36g ) $$\begin{eqnarray}\displaystyle \overline{h}_{j}(t+\unicode[STIX]{x1D6E5}t) & = & \displaystyle \,\overline{h}_{j}^{\mathit{III}},\end{eqnarray}$$

(5.36h ) $$\begin{eqnarray}\displaystyle \overline{q}_{j}(t+\unicode[STIX]{x1D6E5}t) & = & \displaystyle \displaystyle \frac{\,\overline{q}_{j}^{\,\mathit{III}}-(\unicode[STIX]{x1D6E5}t)^{2}{\mathcal{L}}^{(2)}[\,\overline{\boldsymbol{U}}^{\mathit{III}}]_{j}\,{\mathcal{M}}(\,\overline{\boldsymbol{U}}_{j}^{\mathit{III}})}{1+(\unicode[STIX]{x1D6E5}t\,{\mathcal{M}}(\,\overline{\boldsymbol{U}}_{j}^{\mathit{III}}))^{2}},\end{eqnarray}$$

$$\begin{eqnarray}\overline{\boldsymbol{U}}^{I}=(\,\overline{h}^{I},\,\overline{q}^{\,I})^{\top },\quad \,\overline{\boldsymbol{U}}^{\mathit{II}}=(\,\overline{h}^{\mathit{II}},\,\overline{q}^{\,\mathit{II}})^{\top }\quad \text{and}\quad \,\overline{\boldsymbol{U}}^{\mathit{III}}=(\,\overline{h}^{\mathit{III}},\,\overline{q}^{\,\mathit{III}})^{\top }.\end{eqnarray}$$

As has been proved in Chertock et al. (Reference Chertock, Cui, Kurganov and Tong2015a ), the fully discrete scheme (5.36) is well-balanced (in the sense that it is capable of preserving both ‘lake at rest’ (5.25) and ‘slanted surface’ (5.30) steady states) and positivity-preserving. The latter is, in fact, enforced by implementing an adaptive time-step control similar to the one described in Section 5.1.3.

6.1 Well-balanced path-conservative central-upwind schemes

In this section we consider the slightly different non-conservative system

(6.12) $$\begin{eqnarray}\boldsymbol{U}_{t}+\boldsymbol{F}(\boldsymbol{U},B)_{x}={\mathcal{N}}(\boldsymbol{U},B)\boldsymbol{U}_{x}+\boldsymbol{S}(\boldsymbol{U})B_{x},\end{eqnarray}$$

where $B=B(x)$ is a given piecewise smooth function with a finite number of discontinuities. In such a case, the term $\boldsymbol{S}(\boldsymbol{U})B_{x}$ on the right-hand side of (6.12) may represent a geometric source term appearing, for example, in the Saint-Venant system (2.3) or the two-layer shallow-water system (7.4), which will be considered in Section 7.2.

It is possible to apply the above path-conservative central-upwind scheme to the system (6.12). To this end, we add the $(N+1)$ st equation $B_{t}=0$ to (6.12), introduce the extended vector of unknowns $\boldsymbol{V}:=(\boldsymbol{U}^{T},B)^{T}\in \mathbb{R}^{N+1}$ , and rewrite the system (6.12) in the following quasilinear form:

(6.13)

The path-conservative central-upwind scheme (6.9), (6.2), (6.10) can now be directly applied to the system (6.13). However, the resulting scheme will have two major drawbacks. First, the numerical diffusion present in the path-conservative central-upwind scheme will, in general, affect the last equation so that the computed $B$ will not stay time-independent. Second, the scheme will (most probably) not be well-balanced, in the sense that it will not be designed to preserve steady-state solutions of (6.12).

In order to overcome the first of the above difficulties, we apply the path-conservative central-upwind scheme to the first $N$ equations of the system (6.13) only. This results in

(6.14) $$\begin{eqnarray}\displaystyle \frac{\text{d}}{\text{d}t}\,\overline{\boldsymbol{U}}_{j}=-\displaystyle \frac{1}{\unicode[STIX]{x1D6E5}x}(\boldsymbol{D}_{j-1/2}^{+}+\boldsymbol{D}_{j+1/2}^{-}+\boldsymbol{F}(\boldsymbol{V}_{j+1/2}^{-})-\boldsymbol{F}(\boldsymbol{V}_{j-1/2}^{+})-\boldsymbol{{\mathcal{N}}}_{j}-\boldsymbol{S}_{j}),\end{eqnarray}$$

where

(6.15) $$\begin{eqnarray}\displaystyle \boldsymbol{D}_{j+1/2}^{\pm } & = & \displaystyle \displaystyle \frac{1\pm \unicode[STIX]{x1D6FC}_{1}^{j+1/2}}{2}(\boldsymbol{F}(\boldsymbol{V}_{j+1/2}^{+})-\boldsymbol{F}(\boldsymbol{V}_{j+1/2}^{-})-\boldsymbol{{\mathcal{N}}}_{\unicode[STIX]{x1D733},j+1/2}-\boldsymbol{S}_{\unicode[STIX]{x1D733},j+1/2})\nonumber\\ \displaystyle & & \displaystyle \qquad \pm \displaystyle \frac{\unicode[STIX]{x1D6FC}_{0}^{j+1/2}}{2}(\boldsymbol{U}_{j+1/2}^{+}-\boldsymbol{U}_{j+1/2}^{-})\end{eqnarray}$$

and

(6.16a ) $$\begin{eqnarray}\displaystyle \boldsymbol{{\mathcal{N}}}_{j} & := & \displaystyle \int _{C_{j}}{\mathcal{N}}(\boldsymbol{{\mathcal{P}}}_{j}(x))\biggl(\displaystyle \frac{\text{d}P_{j}^{(1)}(x)}{\text{d}x},\ldots ,\displaystyle \frac{\text{d}P_{j}^{(N)}(x)}{\text{d}x}\biggr)^{\!\top }\,\text{d}x,\end{eqnarray}$$

(6.16b ) $$\begin{eqnarray}\displaystyle \boldsymbol{{\mathcal{N}}}_{\unicode[STIX]{x1D733},j+1/2} & := & \displaystyle \int _{0}^{1}{\mathcal{N}}(\unicode[STIX]{x1D733}_{j+1/2}(s))\biggl(\displaystyle \frac{\text{d}\unicode[STIX]{x1D6F9}_{j+1/2}^{(1)}}{\text{d}s},\ldots ,\displaystyle \frac{\text{d}\unicode[STIX]{x1D6F9}_{j+1/2}^{(N)}}{\text{d}s}\biggr)^{\!\top }\,\text{d}s,\end{eqnarray}$$

(6.16c ) $$\begin{eqnarray}\displaystyle \boldsymbol{S}_{j} & := & \displaystyle \int _{C_{j}}\boldsymbol{S}(\boldsymbol{{\mathcal{P}}}_{j}(x))\,\displaystyle \frac{\text{d}P_{j}^{(N+1)}(x)}{\text{d}x}\,\text{d}x,\end{eqnarray}$$

(6.16d ) $$\begin{eqnarray}\displaystyle \boldsymbol{S}_{\unicode[STIX]{x1D733},j+1/2} & := & \displaystyle \int _{0}^{1}\boldsymbol{S}(\unicode[STIX]{x1D733}_{j+1/2}(s))\,\displaystyle \frac{\text{d}\unicode[STIX]{x1D6F9}_{j+1/2}^{(N+1)}}{\text{d}s}\,\text{d}s.\end{eqnarray}$$

$\boldsymbol{V}$

$$\begin{eqnarray}\widetilde{\boldsymbol{V}}(x)=\mathop{\sum }_{j}\boldsymbol{{\mathcal{P}}}_{j}(x)\unicode[STIX]{x1D712}_{j}(x),\quad \boldsymbol{{\mathcal{P}}}_{j}=(P_{j}^{(1)},\ldots ,P_{j}^{(N)},P_{j}^{(N+1)})^{\top }\end{eqnarray}$$

and

$$\begin{eqnarray}\boldsymbol{V}_{j+1/2}^{-}=\boldsymbol{{\mathcal{P}}}_{j}(x_{j+1/2}),\quad \boldsymbol{V}_{j+1/2}^{+}=\boldsymbol{{\mathcal{P}}}_{j+1}(x_{j+1/2}),\end{eqnarray}$$

respectively, a smooth path

$$\begin{eqnarray}\unicode[STIX]{x1D733}_{j+1/2}(s)=(\unicode[STIX]{x1D6F9}_{j+1/2}^{(1)},\ldots ,\unicode[STIX]{x1D6F9}_{j+1/2}^{(N)},\unicode[STIX]{x1D6F9}_{j+1/2}^{(N+1)})^{\top }:=\unicode[STIX]{x1D733}(s;\boldsymbol{V}_{j+1/2}^{-},\boldsymbol{V}_{j+1/2}^{+})\end{eqnarray}$$

now connects the states $\boldsymbol{V}_{j+1/2}^{-}$ and $\boldsymbol{V}_{j+1/2}^{+}$ , that is,

$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D733}:[0,1]\times \mathbb{R}^{N+1}\times \mathbb{R}^{N+1}\rightarrow \mathbb{R}^{N+1}\quad \text{such that}\nonumber\\ \displaystyle & & \displaystyle \unicode[STIX]{x1D733}(0;\boldsymbol{V}_{j+1/2}^{-},\boldsymbol{V}_{j+1/2}^{+})=\boldsymbol{V}_{j+1/2}^{-},\quad \unicode[STIX]{x1D733}(1;\boldsymbol{V}_{j+1/2}^{-},\boldsymbol{V}_{j+1/2}^{+})=\boldsymbol{V}_{j+1/2}^{+},\nonumber\end{eqnarray}$$

and the one-sided local speeds are calculated using the largest ( $\unicode[STIX]{x1D706}_{N}$ ) and smallest ( $\unicode[STIX]{x1D706}_{1}$ ) eigenvalues of

$$\begin{eqnarray}{\mathcal{A}}(\boldsymbol{V})=\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{F}}{\unicode[STIX]{x2202}\boldsymbol{U}}(\boldsymbol{V})-{\mathcal{N}}(\boldsymbol{V}).\end{eqnarray}$$

Since the obtained scheme (6.14)–(6.16) is not guaranteed to preserve steady-state solutions of (6.12), it has to be modified further. In order to construct a well-balanced path-conservative central-upwind scheme, we follow the idea presented in Castro, Pardo, Parés and Toro (Reference Castro, Pardo, Parés and Toro2010), Castro Díaz and Fernández-Nieto (Reference Castro Díaz and Fernández-Nieto2012) and Parés and Castro (Reference Parés and Castro2004), and add an additional term to $\boldsymbol{D}_{j+1/2}^{\pm }$ so that (6.15) is replaced with

(6.17) $$\begin{eqnarray}\displaystyle \boldsymbol{D}_{j+1/2}^{\pm } & = & \displaystyle \displaystyle \frac{1\pm \unicode[STIX]{x1D6FC}_{1}^{j+1/2}}{2}(\boldsymbol{F}(\boldsymbol{V}_{j+1/2}^{+})-\boldsymbol{F}(\boldsymbol{V}_{j+1/2}^{-})-\boldsymbol{{\mathcal{N}}}_{\unicode[STIX]{x1D733},j+1/2}-\boldsymbol{S}_{\unicode[STIX]{x1D733},j+1/2})\nonumber\\ \displaystyle & & \displaystyle \qquad \pm \displaystyle \frac{\unicode[STIX]{x1D6FC}_{0}^{j+1/2}}{2}(\boldsymbol{U}_{j+1/2}^{+}-\boldsymbol{U}_{j+1/2}^{-}-({\mathcal{A}}_{j+1/2}^{\star })^{-1}\widehat{\boldsymbol{S}}_{\unicode[STIX]{x1D733},j+1/2}).\end{eqnarray}$$

Here, ${\mathcal{A}}_{j+1/2}$ is an approximation of the Jacobian matrix ${\mathcal{A}}(\boldsymbol{V})$ near $x=x_{j+1/2}$ (for example, one may use the Roe matrix (Roe Reference Roe1981), but simpler strategies such as those studied in Masella, Faille and Gallouët (Reference Masella, Faille and Gallouët1999) may work as well), ${\mathcal{A}}_{j+1/2}^{\ast }$ is its projection onto the subset of the state space containing the steady-state solutions to be preserved (for details see Castro et al. Reference Castro, Pardo, Parés and Toro2010; for a particular example see Section 7.2.1), and

(6.18) $$\begin{eqnarray}\widehat{\boldsymbol{S}}_{\unicode[STIX]{x1D733},j+1/2}:=\int _{0}^{1}\widehat{\boldsymbol{S}}(\unicode[STIX]{x1D733}_{j+1/2}(s))\,\displaystyle \frac{\text{d}\unicode[STIX]{x1D6F9}_{j+1/2}^{(N+1)}}{\text{d}s}\,\text{d}s.\end{eqnarray}$$

Finally, the scheme (6.14), (6.17) can be recast in the following form (compare with (6.9), (6.2)):

(6.19) $$\begin{eqnarray}\displaystyle \displaystyle \frac{\text{d}}{\text{d}t}\,\overline{\boldsymbol{U}}_{j} & = & \displaystyle -\displaystyle \frac{1}{\unicode[STIX]{x1D6E5}x}\left[\boldsymbol{{\mathcal{F}}}_{j+1/2}-\boldsymbol{{\mathcal{F}}}_{j-1/2}-\boldsymbol{{\mathcal{N}}}_{j}-\boldsymbol{S}_{j}\right.\nonumber\\ \displaystyle & & \displaystyle \qquad -\displaystyle \frac{a_{j-1/2}^{+}}{a_{j-1/2}^{+}-a_{j-1/2}^{-}}(\boldsymbol{{\mathcal{N}}}_{\unicode[STIX]{x1D733},j-1/2}+\boldsymbol{S}_{\unicode[STIX]{x1D733},j-1/2})\nonumber\\ \displaystyle & & \displaystyle \left.\qquad +\displaystyle \frac{a_{j+1/2}^{-}}{a_{j+1/2}^{+}-a_{j+1/2}^{-}}(\boldsymbol{{\mathcal{N}}}_{\unicode[STIX]{x1D733},j+1/2}+\boldsymbol{S}_{\unicode[STIX]{x1D733},j+1/2})\right],\end{eqnarray}$$

where the numerical flux $\boldsymbol{{\mathcal{F}}}_{j+1/2}$ is given by

(6.20) $$\begin{eqnarray}\displaystyle \boldsymbol{{\mathcal{F}}}_{j+1/2} & = & \displaystyle \displaystyle \frac{a_{j+1/2}^{+}\boldsymbol{F}(\boldsymbol{V}_{j+1/2}^{-})-a_{j+1/2}^{-}\boldsymbol{F}(\boldsymbol{V}_{j+1/2}^{+})}{a_{j+1/2}^{+}-a_{j+1/2}^{-}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\displaystyle \frac{a_{j+1/2}^{+}a_{j+1/2}^{-}}{a_{j+1/2}^{+}-a_{j+1/2}^{-}}[\boldsymbol{U}_{j+1/2}^{+}-\boldsymbol{U}_{j+1/2}^{-}-({\mathcal{A}}_{j+1/2}^{\star })^{-1}\widehat{\boldsymbol{S}}_{\unicode[STIX]{x1D733},j+1/2}],\end{eqnarray}$$

Remark 6.2. Notice that if the original central-upwind flux is used instead of the modified flux given by (6.20), then the scheme will be well-balanced only in the case when

$$\begin{eqnarray}\boldsymbol{U}_{j+1/2}^{+}-\boldsymbol{U}_{j+1/2}^{-}\equiv \mathbf{0}\quad \text{for all}~j\end{eqnarray}$$

at steady states. However, in a generic case, $\boldsymbol{U}_{j+1/2}^{+}-\boldsymbol{U}_{j+1/2}^{-}$ does not necessarily vanish, while the corresponding term appearing on the right-hand side of (6.20),

$$\begin{eqnarray}\boldsymbol{U}_{j+1/2}^{+}-\boldsymbol{U}_{j+1/2}^{-}-({\mathcal{A}}_{j+1/2}^{\star })^{-1}\widehat{\boldsymbol{S}}_{\unicode[STIX]{x1D733},j+1/2},\end{eqnarray}$$

is in fact an approximation of $\unicode[STIX]{x1D6E5}x(\boldsymbol{U}_{x}-{\mathcal{A}}(\boldsymbol{V})^{-1}\widehat{\boldsymbol{S}}(\boldsymbol{V})B_{x})$ across the cell interface and it vanishes at steady-state solutions provided a proper path is selected in the evaluation of $\widehat{\boldsymbol{S}}_{\unicode[STIX]{x1D733},j+1/2}$ in (6.18); see Castro et al. (Reference Castro, Pardo, Parés and Toro2010), Castro Díaz and Fernández-Nieto (Reference Castro Díaz and Fernández-Nieto2012) and Parés and Castro (Reference Parés and Castro2004) for details. This guarantees a perfect balance between the source and flux terms as long as the reconstruction (3.6), (3.7) preserves the steady-state solution.

7.1 Shallow-water models with time-dependent bottom topography

In many practically relevant situations, the bottom topography $B$ is time-dependent due to erosion, sediment transport, dam breaks, floods and submarine landslides; see, for example, Grass (Reference Grass1981), Heinrich (Reference Heinrich1992), Hu, Cao, Pender and Tan (Reference Hu, Cao, Pender and Tan2012), Li and Duffy (Reference Li and Duffy2011), Morales de Luna, Castro Díaz, Parés Madroñal and Fernández Nieto (Reference Morales de Luna, Castro Díaz, Parés Madroñal and Fernández Nieto2009), Murillo and García-Navarro (Reference Murillo and García-Navarro2010), Pritchard and Hogg (Reference Pritchard and Hogg2002), Simpson and Castelltort (Reference Simpson and Castelltort2006), Tingsanchali and Chinnarasri (Reference Tingsanchali and Chinnarasri2001), Watts (Reference Watts2000), Wu and Wang (Reference Wu and Wang2007), Xia, Lin, Falconer and Wang (Reference Xia, Lin, Falconer and Wang2010) and Yue, Cao, Li and Che (Reference Yue, Cao, Li and Che2008). Unfortunately, a straightforward application of the finite-volume methods described in Sections 3 and 4 may lead to very inaccurate results in the case of time-dependent bottom topography. This is due to the fact that in this case, the speed of water surface gravity waves is typically much larger than the speed at which the changes in the bottom topography occur. This imposes a severe stability restriction on the size of time steps, which, in turn, leads to excessive numerical diffusion that affects the computed bottom structure.

The simplest way to model the bottom topography propagation was proposed in Exner (Reference Exner1920). According to this approach, in the 2-D case, the bottom topography function satisfies the following equation:

(7.1) $$\begin{eqnarray}B_{t}+\unicode[STIX]{x1D701}q^{B}(h,u,v)_{x}+\unicode[STIX]{x1D701}p^{B}(h,u,v)_{y}=0,\end{eqnarray}$$

where $\unicode[STIX]{x1D701}=1/(1-\unicode[STIX]{x1D6FE})$ is a constant with $\unicode[STIX]{x1D6FE}$ representing the porosity of the sediment layer. The sediment $x$ - and $y$ -discharges, $q^{B}$ and $p^{B}$ , depend on various water and sediment properties. The simplest bottom topography transport fluxes, which were proposed in Grass (Reference Grass1981), are

(7.2) $$\begin{eqnarray}q^{B}=Au(u^{2}+v^{2})^{(m-1)/2},\quad p^{B}=Av(u^{2}+v^{2})^{(m-1)/2},\end{eqnarray}$$

where $A\in [0,1]$ is a non-dimensional constant which accounts for the effects of sediment grain size and kinematic viscosity. The values of $A$ and a constant $1\leq m\leq 4$ are often obtained from the experimental data.

Efficient, accurate and robust numerical methods for the shallow-water system (1.1), (7.1), (7.2) and its 1-D version, the Saint-Venant system (2.3) coupled with the 1-D Exner equation,

(7.3) $$\begin{eqnarray}B_{t}+\displaystyle \frac{A}{1-\unicode[STIX]{x1D6FE}}(u^{m})_{x}=0,\end{eqnarray}$$

have recently been proposed in Bernstein, Chertock, Kurganov and Wu (Reference Bernstein, Chertock, Kurganov and Wu2018) and will now be reviewed briefly.

In principle, one can apply a Godunov-type central or central-upwind scheme to the system (2.3), (7.3) in a straightforward manner. This, however, will lead to excessive smearing of the computed $B$ , as demonstrated in numerical examples reported in Bernstein et al. (Reference Bernstein, Chertock, Kurganov and Wu2018). In order to understand this phenomenon, we first study the Jacobian of the system (2.3), (7.3), which has three real eigenvalues $\unicode[STIX]{x1D706}_{1}<\unicode[STIX]{x1D706}_{2}<\unicode[STIX]{x1D706}_{3}$ . As shown in Bernstein et al. (Reference Bernstein, Chertock, Kurganov and Wu2018), $\unicode[STIX]{x1D706}_{1}$ and $\unicode[STIX]{x1D706}_{3}$ are close to the eigenvalues of the original 1-D Saint-Venant system (2.3), while $\unicode[STIX]{x1D706}_{2}$ is close to zero and typically $|\unicode[STIX]{x1D706}_{2}|\ll \max (|\unicode[STIX]{x1D706}_{1}|,|\unicode[STIX]{x1D706}_{3}|)$ . This corresponds to the bottom topography movement being much slower than the surface wave propagation, and it is well known that slow waves require a special treatment to be accurately resolved numerically.

In order to develop efficient and accurate numerical methods for the system (2.3), (7.3), we apply the second-order Strang operator splitting method (see e.g. Jia and Li Reference Jia and Li2011, Marchuk Reference Marchuk1990, Strang Reference Strang1968, Vabishchevich Reference Vabishchevich2014): we split the Saint-Venant system (2.3) from the Exner equation (7.3). This allows one to treat slow and fast waves in a different manner and using different sizes of time steps. The size of splitting time steps is taken to be proportional to $1/|\unicode[STIX]{x1D706}_{2}|$ . We then follow the approach that was developed in the framework of the fast explicit operator splitting method, which we derived in Chertock, Doering, Kashdan and Kurganov (Reference Chertock, Doering, Kashdan and Kurganov2010), Chertock, Kashdan and Kurganov (Reference Chertock, Kashdan, Kurganov, Benzoni-Gavage and Serre2008), Chertock and Kurganov (Reference Chertock and Kurganov2009) and Chertock, Kurganov and Petrova (Reference Chertock, Kurganov, Petrova and Benkhaldoun2005, Reference Chertock, Kurganov and Petrova2009): each Saint-Venant splitting sub-step consists of several smaller central-upwind steps. In this way we ensure the stability of the Saint-Venant sub-steps, while large Exner splitting sub-steps prevent excessive numerical dissipation, which may severely affect the resolution of the bottom topography, especially in the case when $B$ is discontinuous.

We note that at the Saint-Venant splitting sub-steps we ‘freeze’ the bottom topography, while at the Exner splitting sub-steps we ‘freeze’ the rest of the solution components.

Another difficulty related to the implementation of the central-upwind scheme is the fact that, as we have explained in Section 5.1.1, the evolution of the Saint-Venant solution uses the continuous piecewise linear interpolant of $B$ given in (5.6), which requires the point values of $B$ at the cell interfaces $x=x_{j+1/2}$ . We therefore solve the Exner equation (7.3) on a staggered grid, that is, we evolve the point values $B_{j+1/2}$ rather than $B_{j}$ or the cell averages $\overline{B}_{j}$ . To this end, we need to project the velocities $u_{j}=\,\overline{q}_{j}/\,\overline{h}_{j}$ onto the staggered grid. This can be done in several different ways. We proceed as in Jiang et al. (Reference Jiang, Levy, Lin, Osher and Tadmor1998): reconstruct a piecewise linear interpolant

$$\begin{eqnarray}\widetilde{u}(x)=u_{j}+(u_{x})_{j}(x-x_{j}),\quad x\in C_{j}\end{eqnarray}$$

using a certain nonlinear limiter to evaluate $(u_{x})_{j}$ , then integrate it to obtain

$$\begin{eqnarray}\overline{u}_{j+1/2}=\int _{x_{j}}^{x_{j+1}}\widetilde{u}(x)\,\text{d}x=\displaystyle \frac{1}{2}(u_{j}+u_{j+1})+\displaystyle \frac{\unicode[STIX]{x1D6E5}x}{8}((u_{x})_{j}-(u_{x})_{j+1}).\end{eqnarray}$$

Equipped with the staggered grid data, $\{B_{j+1/2}\}$ and $\{\,\overline{u}_{j+1/2}\}$ , we apply the central-upwind scheme to the Exner equation (7.3) using the local speeds proportional to $\unicode[STIX]{x1D706}_{2}$ , which is small and this guarantees that no excessive numerical diffusion is present at the central-upwind discretization of (7.3); see Bernstein et al. (Reference Bernstein, Chertock, Kurganov and Wu2018) for details.

7.2 Two-layer shallow-water systems

A system of two-layer shallow-water equations, which models oceanographic water flows in straights and channels (see e.g. Castro, Macías and Parés Reference Castro, Macías and Parés2001, Macías, Pares and Castro Reference Macías, Pares and Castro1999), is conceptually more complicated than the single-layer Saint-Venant system since it contains non-conservative interlayer exchange terms. In the 1-D case the system reads as

(7.4a ) $$\begin{eqnarray}\displaystyle (h_{1})_{t}+(q_{1})_{x} & = & \displaystyle 0,\end{eqnarray}$$

(7.4b ) $$\begin{eqnarray}\displaystyle (q_{1})_{t}+\biggl(h_{1}u_{1}^{2}+\displaystyle \frac{1}{2}gh_{1}^{2}\biggr)_{x} & = & \displaystyle -gh_{1}(B+h_{2})_{x},\end{eqnarray}$$

(7.4c ) $$\begin{eqnarray}\displaystyle (h_{2})_{t}+(q_{2})_{x} & = & \displaystyle 0,\end{eqnarray}$$

(7.4d ) $$\begin{eqnarray}\displaystyle (q_{2})_{t}+\biggl(h_{2}u_{2}^{2}+\displaystyle \frac{1}{2}gh_{2}^{2}\biggr)_{x} & = & \displaystyle -gh_{2}(B+rh_{1})_{x},\end{eqnarray}$$

$h_{1}$

$h_{2}$

$u_{i}$

$q_{i}=h_{i}u_{i}$

$i=1,2$

$r\leq 1$

(7.5a ) $$\begin{eqnarray}\displaystyle (h_{1})_{t}+(q_{1}^{x})_{x}+(q_{1}^{y})_{y} & = & \displaystyle 0,\end{eqnarray}$$

(7.5b ) $$\begin{eqnarray}\displaystyle (q_{1}^{x})_{t}+\biggl(h_{1}u_{1}^{2}+\displaystyle \frac{1}{2}gh_{1}^{2}\biggr)_{x}+(h_{1}u_{1}v_{1})_{y} & = & \displaystyle -gh_{1}(B+h_{2})_{x},\end{eqnarray}$$

(7.5c ) $$\begin{eqnarray}\displaystyle (q_{1}^{y})_{t}+(h_{1}u_{1}v_{1})_{x}+\biggl(h_{1}v_{1}^{2}+\displaystyle \frac{1}{2}gh_{1}^{2}\biggr)_{y} & = & \displaystyle -gh_{1}(B+h_{2})_{y},\end{eqnarray}$$

(7.5d ) $$\begin{eqnarray}\displaystyle (h_{2})_{t}+(q_{2}^{x})_{x}+(q_{2}^{y})_{y} & = & \displaystyle 0,\end{eqnarray}$$

(7.5e ) $$\begin{eqnarray}\displaystyle (q_{2}^{x})_{t}+\biggl(h_{2}u_{2}^{2}+\displaystyle \frac{1}{2}gh_{2}^{2}\biggr)_{x}+(h_{2}u_{2}v_{2})_{y} & = & \displaystyle -gh_{2}(B+rh_{1})_{x},\end{eqnarray}$$

(7.5f ) $$\begin{eqnarray}\displaystyle (q_{2}^{y})_{t}+(h_{2}u_{2}v_{2})_{x}+\biggl(h_{2}v_{2}^{2}+\displaystyle \frac{1}{2}gh_{2}^{2}\biggr)_{y} & = & \displaystyle -gh_{2}(B+rh_{1})_{y},\end{eqnarray}$$

$u_{i}$

$q_{i}^{x}=h_{i}u_{i}$

$i=1,2$

$x$

$v_{i}$

$q_{i}^{y}=h_{i}v_{i}$

$i=1,2$

$y$

The two-layer shallow-water systems (7.4) and (7.5) have been extensively studied and a number of numerical methods has been developed; see, for example, Abgrall and Karni (Reference Abgrall and Karni2009), Bouchut and Morales de Luna (Reference Bouchut and Morales de Luna2008), Bouchut and Zeitlin (Reference Bouchut and Zeitlin2010), Castro et al. (Reference Castro, Macías and Parés2001), Castro et al. (Reference Castro, LeFloch, Muñoz-Ruiz and Parés2008), Castro et al. (Reference Castro, Macías, Parés, García-Rodríguez and Vázquez-Cendón2004), Castro Díaz, Chacón Rebollo, Fernández-Nieto and Pares (Reference Castro Díaz, Chacón Rebollo, Fernández-Nieto and Pares2007) and Macías et al. (Reference Macías, Pares and Castro1999). Unfortunately, most of these methods are not sufficiently robust since the computed solution heavily depends on the way the non-conservative product terms on the right-hand side of (7.4) and (7.5) are discretized.

In Kurganov and Petrova (Reference Kurganov and Petrova2009), we have proposed a special way to treat these terms and developed a semi-discrete central-upwind scheme for (7.4) and (7.5). This method is based on the following reformulation of the two-layer shallow-water system. For simplicity, we will consider the 1-D system (7.4) and rewrite it in terms of the equilibrium variables $h_{1}$ , $q_{1}$ , $w:=h_{2}+B$ and $q_{2}$ (notice that at ‘lake at rest’ steady states $h_{1}\equiv \text{const.}$ , $w\equiv \text{const.}$ , $q_{1}\equiv q_{2}\equiv 0$ ) as

(7.6a ) $$\begin{eqnarray}\displaystyle (h_{1})_{t}+(q_{1})_{x} & = & \displaystyle 0,\end{eqnarray}$$

(7.6b ) $$\begin{eqnarray}\displaystyle (q_{1})_{t}+\biggl(\displaystyle \frac{q_{1}^{2}}{h_{1}}+g\unicode[STIX]{x1D700}h_{1}\biggr)_{x} & = & \displaystyle g\unicode[STIX]{x1D700}(h_{1})_{x},\end{eqnarray}$$

(7.6c ) $$\begin{eqnarray}\displaystyle w_{t}+(q_{2})_{x} & = & \displaystyle 0,\end{eqnarray}$$

(7.6d ) $$\begin{eqnarray}\displaystyle (q_{2})_{t}+\biggl(\displaystyle \frac{q_{2}^{2}}{w-B}+\displaystyle \frac{g}{2}w^{2}-\displaystyle \frac{g}{2}rh_{1}^{2}-gB\widehat{\unicode[STIX]{x1D700}}\biggr)_{x} & = & \displaystyle -g\widehat{\unicode[STIX]{x1D700}}B_{x}-gr\unicode[STIX]{x1D700}(h_{1})_{x},\end{eqnarray}$$

$\unicode[STIX]{x1D700}=h_{1}+h_{2}+B=h_{1}+w$

$\widehat{\unicode[STIX]{x1D700}}:=rh_{1}+w$

Figure 7.1.

Set-up for the two-layer shallow-water system (7.6).

Notice that while the two-layer shallow-water system (7.6) is equivalent (for smooth solutions) to the original two-layer system (7.4), it is suitable for a ‘favourable’ treatment of the non-conservative products. Our approach uses the fact that in the relevant applications, fluctuations of the total water level $\unicode[STIX]{x1D700}$ are relatively small, that is, after choosing a proper coordinate system, $\unicode[STIX]{x1D700}\sim 0$ ; see Figure 7.1. Thus we reduce as much as possible the ‘influence’ of the particular choice of the non-conservative product discretization as the non-conservative product terms on the right-hand side of (7.6) are proportional to $\unicode[STIX]{x1D700}$ .

The rewritten systems are then numerically solved by a two-layer version of the second-order well-balanced positivity-preserving central-upwind scheme presented in Sections 5.1.1 and 5.1.3. To this end, we (i) replace the bottom with its piecewise linear approximation $\widetilde{B}$ ; (ii) correct the reconstruction of $w$ to ensure that $\widetilde{w}\geq \widetilde{B}$ ; (iii) regularize the computed velocities to avoid division by $h_{i}\sim 0$ ; (iv) use the well-balanced quadrature for the geometric source term. The only difficulty with the implementation of the central-upwind scheme is related to the fact that one cannot find analytical expressions for the Jacobian eigenvalues needed to evaluate one-sided local speeds in (3.14). Indeed, as one can easily see, the eigenvalues are solutions of the following algebraic equation:

(7.7) $$\begin{eqnarray}(\unicode[STIX]{x1D706}^{2}-2u_{1}\unicode[STIX]{x1D706}+u_{1}^{2}-gh_{1})(\unicode[STIX]{x1D706}^{2}-2u_{2}\unicode[STIX]{x1D706}+u_{2}^{2}-gh_{2})=g^{2}\widehat{h}_{1}h_{2}.\end{eqnarray}$$

Although this equation cannot be explicitly solved, one can easily obtain an upper bound on the largest eigenvalue and a lower bound on the smallest eigenvalue using the Lagrange theorem; see, for example, Lagrange (Reference Lagrange1798) and Mignotte and Stefanescu (Reference Mignotte and Stefanescu2002).

Remark 7.1. One can show that equation (7.7) may have either four or only two real solutions depending on the values of $h_{1}$ , $h_{2}$ , $u_{1}$ , $u_{1}$ and $r$ . This means that the two-layer shallow-water system (7.4) is only conditionally hyperbolic and, as observed in the literature, in the non-hyperbolic regime, its solution typically develops instabilities.

To study the hyperbolicity regions of (7.4), one can examine the first-order approximation of its Jacobian eigenvalues (see e.g. Castro et al. Reference Castro, Macías and Parés2001, Schijf and Schonfeld Reference Schijf and Schonfeld1953), which is given by

(7.8a ) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D706}_{1,2}(h_{1},u_{1},h_{2},u_{2})\approx U_{m}\pm \sqrt{g(h_{1}+h_{2})},\end{eqnarray}$$

(7.8b ) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D706}_{3,4}(h_{1},u_{1},h_{2},u_{2})\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad \approx U_{c}\pm \sqrt{(1-r)g\displaystyle \frac{h_{1}h_{2}}{h_{1}+h_{2}}\biggl(1-\displaystyle \frac{(u_{2}-u_{1})^{2}}{(1-r)g(h_{1}+h_{2})}\biggr)},\end{eqnarray}$$

where

(7.9) $$\begin{eqnarray}U_{m}=\displaystyle \frac{h_{1}u_{1}+h_{2}u_{2}}{h_{1}+h_{2}},\quad U_{c}=\displaystyle \frac{h_{1}u_{2}+h_{2}u_{1}}{h_{1}+h_{2}}.\end{eqnarray}$$

From (7.8), (7.9) one expects that the two-layer shallow-water system (7.4) is hyperbolic as long as

$$\begin{eqnarray}(u_{2}-u_{1})^{2}<(1-r)g(h_{1}+h_{2}),\end{eqnarray}$$

and thus the hyperbolicity condition for the two-layer shallow-water system (7.4) depends on the relationship between $(u_{2}-u_{1})^{2}$ and $(1-r)g(h_{1}+h_{2})$ .

A detailed description of the resulting central-upwind scheme together with an illustration of its performance can be found in Kurganov and Petrova (Reference Kurganov and Petrova2009). However, in examples in which the surface waves are not small, the method from Kurganov and Petrova (Reference Kurganov and Petrova2009) may fail. Therefore, in Castro Díaz et al. (Reference Castro Díaz, Kurganov and Morales de Luna2018) we have developed a well-balanced path-conservative central-upwind scheme, which seems to be a robust numerical method for two-layer shallow-water system (7.4).

7.2.1 Well-balanced path-conservative central-upwind scheme for the two-layer shallow-water system

In order to design a well-balanced path-conservative central-upwind scheme for the two-layer shallow-water system (7.6), we proceed exactly as in Section 6.1 and add the fifth equation $B_{t}=0$ to (7.6). We then rewrite the system (7.6) in the vector form (6.13) with

$$\begin{eqnarray}\displaystyle \boldsymbol{U}=\left(\begin{array}{@{}c@{}}h_{1}\\[2.0pt] q_{1}\\[2.0pt] w\\[2.0pt] q_{2}\end{array}\right),~\boldsymbol{V}=\left(\begin{array}{@{}c@{}}h_{1}\\[2.0pt] q_{1}\\[2.0pt] w\\[2.0pt] q_{2}\\[2.0pt] B\end{array}\right),~\boldsymbol{F}(\boldsymbol{V})=\left(\begin{array}{@{}c@{}}q_{1}\\[2.0pt] {\textstyle \frac{q_{1}^{2}}{h_{1}}}+g(h_{1}+w)h_{1}\\[2.0pt] q_{2}\\[2.0pt] {\textstyle \frac{q_{2}^{2}}{w-B}}+{\textstyle \frac{g}{2}}(w^{2}-rh_{1}^{2})-g(rh_{1}+w)B\end{array}\right), & & \displaystyle \nonumber\\ \displaystyle B(\boldsymbol{V})=\left(\begin{array}{@{}cccc@{}}0 & 0 & 0 & 0\\[2.0pt] g(h_{1}+w) & 0 & 0 & 0\\[2.0pt] 0 & 0 & 0 & 0\\[2.0pt] -gr(h_{1}+w) & 0 & 0 & 0\end{array}\right)\quad \text{and}\quad \boldsymbol{S}(\boldsymbol{U})=\left(\begin{array}{@{}c@{}}0\\[2.0pt] 0\\[2.0pt] 0\\[2.0pt] -g(rh_{1}+w)\end{array}\right). & & \displaystyle \nonumber\end{eqnarray}$$

Notice that at the ‘lake at rest’ steady states,

(7.10) $$\begin{eqnarray}q_{1}\equiv q_{2}\equiv 0,\quad h_{1}\equiv \text{const.},\quad w=h_{2}+B\equiv \text{const.},\end{eqnarray}$$

that is, $\boldsymbol{U}\equiv \mathbf{const.}$ , and thus

$$\begin{eqnarray}\boldsymbol{U}_{j+1/2}^{+}-\boldsymbol{U}_{j+1/2}^{-}\equiv \mathbf{0}\quad \text{for all}~j,\end{eqnarray}$$

provided the piecewise polynomial reconstruction (3.6) exactly recovers the constant states (7.10). Therefore, a well-balanced central-upwind scheme will be obtained with the help of the simplified central-upwind flux (6.2) instead of a more complicated flux (6.20) derived in Section 6.1.

The path-conservative central-upwind scheme (6.19), (6.2), (6.16) is now applied to the two-layer shallow-water system in a straightforward manner. We use the piecewise linear reconstruction

$$\begin{eqnarray}\boldsymbol{{\mathcal{P}}}_{j}(x)=\,\overline{\boldsymbol{V}}_{j}+(\boldsymbol{V}_{x})_{j}(x-x_{j}),\end{eqnarray}$$

which leads to the following expressions for $\boldsymbol{{\mathcal{N}}}_{j}$ and $\boldsymbol{S}_{j}$ :

$$\begin{eqnarray}\displaystyle \boldsymbol{{\mathcal{N}}}_{j} & = & \displaystyle \left(\begin{array}{@{}c@{}}0\\[2.0pt] \displaystyle g((\,\overline{h_{1}})_{j}+\,\overline{w}_{j})((h_{1})_{j+1/2}^{-}-(h_{1})_{j-1/2}^{+})\\[5.0pt] 0\\[2.0pt] -gr((\,\overline{h_{1}})_{j}+\,\overline{w}_{j})((h_{1})_{j+1/2}^{-}-(h_{1})_{j-1/2}^{+})\end{array}\right),\nonumber\\ \displaystyle \boldsymbol{S}_{j} & = & \displaystyle \left(\begin{array}{@{}c@{}}0\\[2.0pt] 0\\[2.0pt] 0\\[2.0pt] -g(r(\,\overline{h_{1}})_{j}+\,\overline{w}_{j})(B_{j+1/2}^{-}-B_{j-1/2}^{+})\end{array}\right),\nonumber\end{eqnarray}$$

and the linear segment path

$$\begin{eqnarray}\unicode[STIX]{x1D733}_{j+1/2}(s)=\boldsymbol{V}_{j+1/2}^{-}+s(\boldsymbol{V}_{j+1/2}^{+}-\boldsymbol{V}_{j+1/2}^{-}),\end{eqnarray}$$

which results in

$$\begin{eqnarray}\boldsymbol{{\mathcal{N}}}_{\unicode[STIX]{x1D733},j+1/2}=\left(\begin{array}{@{}c@{}}0\\[2.0pt] g\unicode[STIX]{x1D719}_{j+1/2}/2\\[2.0pt] 0\\[2.0pt] -gr\unicode[STIX]{x1D719}_{j+1/2}/2\end{array}\right),\end{eqnarray}$$

where

$$\begin{eqnarray}\unicode[STIX]{x1D719}_{j+1/2}=((h_{1})_{j+1/2}^{+}+w_{j+1/2}^{+}+(h_{1})_{j+1/2}^{-}+w_{j+1/2}^{-})((h_{1})_{j+1/2}^{+}-(h_{1})_{j+1/2}^{-}),\end{eqnarray}$$

and

$$\begin{eqnarray}\boldsymbol{S}_{\unicode[STIX]{x1D733},j+1/2}=\left(\begin{array}{@{}c@{}}0\\[2.0pt] 0\\[2.0pt] 0\\[2.0pt] -g\unicode[STIX]{x1D713}_{j+1/2}/2\end{array}\right),\end{eqnarray}$$

where

$$\begin{eqnarray}\unicode[STIX]{x1D713}_{j+1/2}=(r(h_{1})_{j+1/2}^{+}+w_{j+1/2}^{+}+r(h_{1})_{j+1/2}^{-}+w_{j+1/2}^{-})(B_{j+1/2}^{+}-B_{j+1/2}^{-}).\end{eqnarray}$$

In order to design the scheme (6.19), (6.2) for the two-layer system (7.6), we use the matrix

$$\begin{eqnarray}{\mathcal{A}}(\boldsymbol{V})=\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{F}}{\unicode[STIX]{x2202}\boldsymbol{U}}(\boldsymbol{V})-{\mathcal{N}}(\boldsymbol{V})=\left(\begin{array}{@{}cccc@{}}0 & 1 & 0 & 0\\[2.0pt] gh_{1}-u_{1}^{2} & 2u_{1} & gh_{1} & 0\\[2.0pt] 0 & 0 & 0 & 1\\[2.0pt] grh_{2} & 0 & gh_{2}-u_{2}^{2} & 2u_{2}\end{array}\right),\end{eqnarray}$$

where $h_{2}=w-B$ . The one-sided local speeds

$$\begin{eqnarray}\displaystyle a_{j+1/2}^{-} & = & \displaystyle \min \{\unicode[STIX]{x1D706}_{1}({\mathcal{A}}(\boldsymbol{V}_{j+1/2}^{-})),\,\unicode[STIX]{x1D706}_{1}({\mathcal{A}}(\boldsymbol{V}_{j+1/2}^{+})),\,0\},\nonumber\\ \displaystyle a_{j+1/2}^{+} & = & \displaystyle \max \{\unicode[STIX]{x1D706}_{4}({\mathcal{A}}(\boldsymbol{V}_{j+1/2}^{-})),\,\unicode[STIX]{x1D706}_{4}({\mathcal{A}}(\boldsymbol{V}_{j+1/2}^{+})),\,0\}\nonumber\end{eqnarray}$$

are then obtained using the upper/lower bounds on the largest/smallest eigenvalues of ${\mathcal{A}}(\boldsymbol{V})$ using the Lagrange theorem as discussed above.

Remark 7.2. When the well-balanced path-conservative central-upwind scheme is designed, we use a generically discontinuous piecewise polynomial reconstruction of $B$ (which is the fifth component of $\boldsymbol{V}$ ) rather than the continuous piecewise linear one (5.6).