Error Bounds for Fractional Step Methods for Conservation Laws with Source Terms

Abstract

Fractional step methods have been used to approximate solutions of scalar conservation laws with source terms. In this paper, the stability and accuracy of the basic fractional step algorithms are analyzed when these algorithms are used to compute discontinuous solutions of nonhomogeneous scalar conservation laws. The authors show that time-splitting methods for conservation laws with source terms always converge to the unique weak solution satisfying the entropy condition. In particular, it is proved that the $L^1 $ errors in the splitting methods are bounded by $O(\sqrt {\Delta t} )$, where $\Delta t$ is the splitting time step. The $L^1 $ convergence rate of a class of fully discrete splitting methods is also investigated.