On the Numerical Solution of Initial/Boundary-Value Problems in One Space Dimension

Abstract

The numerical solution of initial/boundary-value problems of the form \[ A(u,x,t)u_t + B(u,x,t)u_x = c(u,x,t)\] is considered. Particular emphasis is placed on the solution of problems with large gradients, e.g., shocks and boundary layers. A mesh-selection technique is described that accurately places points in regions where the solution is rapidly changing. This is accomplished by a transformation of the original equations to a new coordinate system. Finite difference solutions for two sample problems are calculated.