Stability theorem and truncation error analysis for the Glimm scheme and for a front tracking method for flows with strong discontinuities

Abstract

AbstractTypical nonlinear wave interaction problems involve strong waves moving through a background of weak disturbance. Previous existence theorems and error analysis for computations are usually restricted to more idealized situations such as small data or single equations. We consider here the problem of a single strong discontinuity interacting with a weak background for general hyperbolic systems of conservation laws. We obtain the stability, consistency theorems and upper bounds of the truncation errors for the Glimm scheme and for a front tracking method. The major error in the Glimm scheme is the error generated by the strong discontinuity. This error is reduced when a front tracking method is applied to follow the location of the strong discontinuity. This demonstrates an advantage of front tracking methods in one‐space dimension.