A family of arbitrarily high-order energy-preserving methods are developed to solve the coupled Schrödinger–Boussinesq (S-B) system. The system is a nonlinear coupled system and satisfies a series of conservation laws. It is often difficult to construct a high-order decoupling numerical algorithm to solve the nonlinear system. In this paper, the original system is first reformulated into an equivalent Hamiltonian system by introducing multiple auxiliary variables. Next, the reformulated system is discretized by the Fourier pseudo-spectral method and the implicit midpoint scheme in the spatial and temporal directions, respectively, and a second-order conservative scheme is obtained. Finally, the scheme is extended to arbitrarily high-order accuracy by means of diagonally implicit symplectic Runge–Kutta methods or composition methods. Rigorous analyses show that the proposed methods are fully decoupled and can precisely conserve the discrete invariants. Numerical results show that the proposed schemes are effective and can be easily extended to other nonlinear partial differential equations.

A Doyle–Fuller–Newman (DFN) model for the charge and discharge of nano-structured lithium iron phosphate (LFP) cathodes is formulated on the basis that lithium transport within the nanoscale LFP electrode particles is much faster than cell discharge, and is therefore not rate limiting. We present some numerical solutions to the model and show that for relevant parameter values, and a variety of C-rates, it is possible for sharp discharge fronts to form and intrude into the electrode from its outer edge(s). These discharge fronts separate regions of fully utilised LFP electrode particles from those that are not. Motivated by this observation an asymptotic solution to the model is sought. The results of the asymptotic analysis of the DFN model lead to a reduced order model, which we term the reaction front model (or RFM). Favourable agreement is shown between solutions to the RFM and the full DFN model in appropriate parameter regimes. The RFM is significantly cheaper to solve than the DFN model, and therefore has the potential to be used in scenarios where computational costs are prohibitive, e.g. in optimisation and parameter estimation problems or in engineering control systems.

We propose two linearly implicit energy-preserving schemes for the complex modified Korteweg–de Vries equation, based on the invariant energy quadratization method. First, a new variable is introduced and a new Hamiltonian system is constructed for this equation. Then the Fourier pseudospectral method is used for the space discretization and the Crank–Nicolson leap-frog schemes for the time discretization. The proposed schemes are linearly implicit, which is only needed to solve a linear system at each time step. The fully discrete schemes can be shown to conserve both mass and energy in the discrete setting. Some numerical examples are also presented to validate the effectiveness of the proposed schemes.

The displacement λ-convexity of a non-standard entropy with respect to a non-local transportation metric in finite state spaces is shown using a gradient flow approach. The constant λ is computed explicitly in terms of a priori estimates of the solution to a finite-difference approximation of a non-linear Fokker–Planck equation. The key idea is to employ a new mean function, which defines the Onsager operator in the gradient flow formulation.

A new Semi-Lagrangian scheme is proposed to discretize the surface convection-diffusion equation. The other involved equations including the the level-set convection equation, the re-initialization equation and the extension equation are also solved by S-L schemes. The S-L method removes both the CFL condition and the stiffness caused by the surface Laplacian, allowing larger time step than the Eulerian method. The method is extended to the block-structured adaptive mesh. Numerical examples are given to demonstrate the efficiency of the S-L method.

This paper is devoted to an extension of the finite-energy condition for extended Runge-Kutta-Nyström (ERKN) integrators and applications to nonlinear wave equations. We begin with an error analysis for the integrators for multi-frequency highly oscillatory systems , where M is positive semi-definite, . The highly oscillatory system is due to the semi-discretisation of conservative, or dissipative, nonlinear wave equations. The structure of such a matrix M and initial conditions are based on particular spatial discretisations. Similarly to the error analysis for Gaustchi-type methods of order two, where a finite-energy condition bounding amplitudes of high oscillations is satisfied by the solution, a finite-energy condition for the semi-discretisation of nonlinear wave equations is introduced and analysed. These ensure that the error bound of ERKN methods is independent of . Since stepsizes are not restricted by frequencies of M, large stepsizes can be employed by our ERKN integrators of arbitrary high order. Numerical experiments provided in this paper have demonstrated that our results are truly promising, and consistent with our analysis and prediction.

In this article, we give a unified theory for constructing boundary layer expansions for discretized transport equations with homogeneous Dirichlet boundary conditions. We exhibit a natural assumption on the discretization under which the numerical solution can be written approximately as a two-scale boundary layer expansion. In particular, this expansion yields discrete semigroup estimates that are compatible with the continuous semigroup estimates in the limit where the space and time steps tend to zero. The novelty of our approach is to cover numerical schemes with arbitrarily many time levels.

In this paper we consider a discontinuous Galerkin discretization of the ideal magnetohydrodynamics (MHD) equations on unstructured meshes, and the divergence free constraint (∇·B =0) of its magnetic field B . We first present two approaches for maintaining the divergence free constraint, namely the approach of a locally divergence free projection inspired by locally divergence free elements [19], and another approach of the divergence cleaning technique given by Dedner et al. [15]. By combining these two approaches we obtain a efficient method at the almost same numerical cost. Finally, numerical experiments are performed to show the capacity and efficiency of the scheme.

Semi-Lagrangian (S-L) methods have no CFL stability constraint, and are more stable than the Eulerian methods. In the literature, the S-L method for the level-set re-initialization equation was complicated, which may be unnecessary. Since the re-initialization procedure is auxiliary, we propose to use the first-order S-L scheme coupled with a projection technique to improve the accuracy at the grid points just adjacent to the interface. Standard second-order S-L method is used for evolving the level-set convection equation. The implementation is simple, including on the block-structured adaptive mesh. The efficiency of the S-L method is demonstrated by extensive numerical examples including passive convection of interfaces with corners/kinks/large deformation under given velocity fields, a geometrical flow with topological changes, simulations of bubble/ droplet dynamics in incompressible two-phase flows. In terms of accuracy it is comparable to the other existing methods.

Novel dimensional splitting techniques are developed for ETD Schemes which are second-order convergent and highly efficient. By using the ETD-Crank-Nicolson scheme we show that the proposed techniques can reduce the computational time for nonlinear reaction-diffusion systems by up to 70%. Numerical tests are performed to empirically validate the superior performance of the splitting methods.

This paper explores the discrete singular convolution method for Hamiltonian PDEs. The differential matrices corresponding to two delta type kernels of the discrete singular convolution are presented analytically, which have the properties of high-order accuracy, bandlimited structure and thus can be excellent candidates for the spatial discretizations for Hamiltonian PDEs. Taking the nonlinear Schrödinger equation and the coupled Schrödinger equations for example, we construct two symplectic integrators combining this kind of differential matrices and appropriate symplectic time integrations, which both have been proved to satisfy the square conservation laws. Comprehensive numerical experiments including comparisons with the central finite difference method, the Fourier pseudospectral method, the wavelet collocation method are given to show the advantages of the new type of symplectic integrators.

It has been known for a long time that the equivariant $2+1$ wave map into the $2$-sphere blows up if the initial data are chosen appropriately. Here, we present numerical evidence for the stability of the blow-up phenomenon under explicit violations of equivariance.