Analysis of the error due to operator splitting in nonlinear systems of hyperbolic conservation laws with source terms

Description

Very often systems of nonlinear hyperbolic conservation laws with source terms are solved by the so called operator splitting, i.e. one alternates between a time step with no source term and one with no transport term. Even if both subproblems are solved exactly the overall scheme may produce wrong shock speeds. For a scalar Riemann problem it has been shown that the local error of the shock location can be considered to consist of two parts: one part introduced by the splitting and another occurring because of smeared-out shock profiles. Numerical examples show that these error-estimates can be used to adapt the stepsize so that the error of the shock location remains sufficiently small. The numerical examples include one- and 2 dimensional systems such as the inviscid reacting compressible Euler equations where the Chapman-Jouguet detonations are computed as well as the one-dimensional combustion model of Majda. The results are collected in the thesis of P. Klingenstein and can also be found in the article by R. Jeltsch and P. Klingenstein to be published in the journal Computing and Visualization in Science, in 1999.

Contacts

Prof. Rolf Jeltsch and Petra Klingenstein.