High Resolution Schemes for Conservation Laws with Locally Varying Time Steps

Abstract

. We develop upwind methods which use limited high resolution corrections in the spatial discretization and local time stepping for forward Euler and second order time discretizations. L 1 stability is proven for both time stepping schemes for problems in one space dimension. These methods are restricted by a local CFL condition rather than the traditional global CFL condition, allowing local time renement to be coupled to local spatial renement. Numerical evidence demonstrates the stability and accuracy of the methods for problems in both one and two space dimensions. Key words. spatially varying time steps, upwinding, conservation laws AMS subject classications. 35L65, 65M12, 65M30 1. Introduction. Hyperbolic conservation laws model a wide range of physically interesting phenomena such as gas dynamics, shallow water ow, and advection of contaminants. Conservative, high resolution methods with explicit time discretization have proven eective in capturing the sharp, moving f...