High Resolution Schemes for Conservation Laws with Locally Varying Time Steps

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^∞ 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 refinement to be coupled with local spatial refinement. Numerical evidence demonstrates the stability and accuracy of the methods for problems in both one and two space dimensions.