First-Order Methods for Systems

First-order monotone numerical methods for non-linear systems of hyperbolic conservation laws are studied. Some of these form the bases for constructing higher-order schemes in successive chapters. Algorithms for one-dimensional PDEs are first presented; these include the Godunov upwind scheme [1] in conjunction with the exact Riemann solver; the Random Choice Method (RCM) of Glimm [2]; Flux-Vector Splitting (FVS) methods; and centred schemes, such as the Lax–Friedrichs [3], Lax–Wendroff [3] and FORCE methods [4, 5].  There follows the finite volume framework for PDEs in multiple space dimensions on unstructured meshes, including definitions for semi-discrete and fully discrete schemes; the rotated Riemann problem from the rotational invariance of the two-dimensional shallow water equations; the intercell numerical flux and determination of edge lengths, normals and cell areas in two space dimensions, in order to completely specify the numerical methods. Some examples are shown, in order to illustrate the performance of selected schemes studied, as compared against exact solutions for a suite of carefully selected test problems.