The Convergence Rate of Godunov Type Schemes

Abstract

Godunov type schemes form a special class of transport projection methods for the approximate solution of nonlinear hyperbolic conservation laws. The authors study the convergence rate of such schemes in the context of scalar conservation laws and show how the question of consistency for Godunov type schemes can be answered solely in terms of the behavior of the associated projection operator. Namely, they prove that ${\textit{Lip}}'$-consistent projections guarantee the ${\textit{Lip}}'$-convergence of the corresponding Godunov scheme, provided the latter is ${\textit{Lip}}^+$-stable. This ${\textit{Lip}}'$-error estimate is then translated into the standard $W^{s,p} $ global error estimates $( - 1 \leq s \leq \frac{1}{p},1 \leq p \leq \infty )$ and finally to a local $L_{{\text{loc}}}^\infty $ convergence rate estimate. These convergence rate estimates are applied to a variety of scalar Godunov type schemes on a uniform grid as well as variable mesh size ones.