24 the solution of the forward time backward space

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: nitial data to determine the solution at (j x n t). These simple arguments lead to the famous Courant, Friedrichs, Lewy Theorem 2, 1] which we'll state in the context of (2.2.1). Theorem 2.2.1. (Courant, Friedrichs, Lewy). A necessary condition for the conver- gence of the solution of a nite-di erence approximation to the solution of (2.2.1) for arbitrary initial data is that the domain of dependence of the nite-di erence approximation contain the domain of dependence of the partial di erential equation (2.2.1). Remark 2. We'll give a formal de nition of convergence in the next section however, informally, convergence implies that the solution of the di erence scheme approaches the 14 Finite Di erence Methods 3 2 U 1 0 −1 −2 −3 4 8 3 6 2 4 1 2 0 t 0 x 1 U 0.5 0 −0.5 −1 4 8 3 6 2 4 1 2 0 t 0 x Figure 2.2.4: Solutions of Example 2.2.2 obtained by the forward time-forward space scheme (2.2.2b) (top) and forward time-backward space scheme (2.2.4) (bottom). Each solution has a Courant number of 1 2. = 2.2. A Kinematic...
View Full Document

This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.

Ask a homework question - tutors are online