Nonlinear Stability

Abstract

The Lax-Wendroff Theorem presented in Chapter 12 does not say anything about whether the method converges, only that if a sequence of approximations converges then the limit is a weak solution. To guarantee convergence, we need some form of stability, just as for linear problems. Unfortunately, the Lax Equivalence Theorem no longer holds and we cannot use the same approach (which relies heavily on linearity) to prove convergence. In this chapter we consider one form of nonlinear stability that allows us to prove convergence results for a wide class of practical methods. So far, this approach has been completely successful only for scalar problems. For general systems of equations with arbitrary initial data no numerical method has been proved to be stable or convergent, although convergence results have been obtained in some special cases (e.g. [20], [50], [53]).