On local conservation of numerical methods for conservation laws

Abstract

In this paper we introduce a definition of the local conservation property for numerical methods solving time dependent conservation laws, which generalizes the classical local conservation definition. The motivation of our definition is the Lax–Wendroff theorem, and thus we prove it for locally conservative numerical schemes per our definition in one and two space dimensions. Several numerical methods, including continuous Galerkin methods and compact schemes, which do not fit the classical local conservation definition, are given as examples of locally conservative methods under our generalized definition.