Introduction

In Chapter 3 we have noted that hyperbolic PDEs are characterized by the presence of real lines along which the highest order derivatives appearing in the differential equations are discontinuous; these lines are the characteristics. Along the characteristics, information propagates for hyperbolic equations. Thus, accurate solution of hyperbolic PDEs is related to the central problem of correctly allowing the signal/disturbance to propagate, as given by the governing PDE. For other classes of differential equations (parabolic or elliptic PDEs) describing equilibrium state of dynamical systems can also support disturbances which can be posed as wave propagation problems. Such wave propagation problems are decided by the dispersion relation relating spatial and temporal scales (as described in Chapter 4 for some mechanical systems) and resultant waves are termed as dispersive waves.

Thus, we discuss about solution methods for hyperbolic PDEs, while focussing on propagating signals and numerical errors. Although, we discuss only about a few handful classical methods of solving hyperbolic PDEs, our attention is equally on the most important problem of signal and error dynamics from a numerical perspective. We would, however, emphasize on signal and error propagation in this chapter, noting that the celebrated error analysis due to von Neumann [41, 53] has been corrected in [259] with respect to a specific example drawn from hyperbolic equation. This forms the second part of this chapter, where the analysis is for identifying mechanisms and sources of numerical error which is independent of which method is being employed.

Introduction

To solve governing equation of motions, we need to resolve all the excited length and time scales. However, even to solve flow past a flat plate, if one takes uniform grid to accommodate the smallest energy carrying length scales, one would be forced to take too many grid points. Numerical solution of the Euler and Navier–Stokes equations for external flow problems requires an outer boundary far away, where some asymptotic boundary conditions apply. This also adds larger requirement on the grid points. Such large problems are poorly convergent and hence, not within the range of the available resources of prevalent high performance computing platforms. This is circumvented by using non-uniform grids. For example, we may decide to take finer grids inside the shear layer (or within the inner layer of a turbulent boundary layer), and take fewer points in the inviscid part of the flow. With nonuniform grids in the physical plane, we usually transform the governing equation in the computational plane, where the spacing is uniform. This also makes writing a code much easier, because of the uniform grid in transformed plane. One needs to write a single code for the transformed plane problem, and for flow past different geometries, one uses the same code provided one can generate a grid mapping from the physical to the transformed plane. Thus, the main solver is grid-independent, and all one needs to do is to generate an appropriate grid transformation either analytically or numerically.