Error Bounds for the Methods of Glimm, Godunov and LeVeque

Abstract

The expected error in $L^1 (R)$ at time T for Glimm’s scheme when applied to a scalar conservation law is bounded by \[\left( {h + \frac{2}{{\sqrt 3 }}\left( {\frac{h}{{\Delta t}}} \right)^{{1 / 2}} } \right)\left\| {u_0 } \right\|_{BV({\bf R})} ,\] where h is the mesh size and $\Delta t$ is the time step. We show that the error in Godunov’s scheme is bounded by the same expression and use this result to analyze a variant of a large time step numerical method of LeVeque that may be viewed as a combination of Godunov’s scheme and a method of Dafermos.