Numerical boundary layers of conservation laws with relaxation extension

Abstract

This paper is concerned with the asymptotic equivalence between scalar conservation law and its relaxing scheme in the presence of boundaries. The main goals are to understand the evolution and structures of numerical boundary layers. We first construct formally the approximate solution up to any order of accuracy by using the matched asymptotic analysis and multiple scale expansions. Next, we prove that the weak numerical boundary layers (those with suitably small strength) are always nonlinearly stable, thus the effects of numerical boundary layers are localized. Finally, we show that the strong numerical boundary layers are nonlinearly stable also which depend crucially on the structures of the numerical boundary layers. The proofs are based weighted energy estimates. Such analysis also gives us some hints to choose boundary conditions in practical computations.