21 - Multidimensional Systems

Summary

In this chapter the high-resolution wave-propagation algorithms developed in Chapter 20 for scalar problems are extended to hyperbolic systems. We start with constant-coefficient linear systems, where the essential ingredients are most easily seen. A Riemann problem is first solved normal to each cell edge (a simple eigendecomposition in the linear case). The resulting waves are used to update cell averages on either side. The addition of correction terms using wave limiters (just as in one dimension) gives high-resolution terms modeling the pure x- and y-derivative terms in the Taylor series expansion (19.5). The crossderivative terms are handled by simple extension of the corner-transport upwind (CTU) idea presented for the advection equation in Sections 20.2 through 20.5. In general this requires solving a second set of Riemann problems transverse to the interface. For a linear system this means performing a second eigendecomposition using the coefficient matrix in the transverse direction. Extending the methods to variable-coefficient or nonlinear systems is then easy, using ideas that are already familiar from one space dimension. The solutions (or approximate solutions) to the more general Riemann problems are used in place of the eigendecompositions, and the method is implemented in a wave-propagation form that applies very generally.

Constant-Coefficient Linear Systems

We again consider the constant-coefficient linear system q_t + Aq_x + Bq_y = 0 discussed in Chapter 19, where in particular the Lax–Wendroff and Godunov methods for this system were presented. The numerical fluxes for these two methods are given by (19.14) and (19.18) respectively.