Stability of Initial-Boundary-Value Schemes

Abstract

Since early in Chapter 1, we have been computing solutions to initial-boundary-value problems. In we included some theory that could be used to prove convergence of schemes for solving initial-boundary-value problems. In Example 2.2.2 we used the definition of convergence to prove the convergence of the basic difference scheme for the heat equation with zero Dirichlet boundary conditions. For the same difference scheme, in Section 2.5.2 we noted that the consistency and stability analyses done earlier in the text along with the Lax Theorem for a bounded domain (Theorem 2.5.3) imply convergence. We also pointed out that we could directly apply the definitions of consistency and stability, and Theorem 2.5.3 to obtain convergence for a hyperbolic scheme.