Multiresolution schemes for conservation laws with viscosity

This paper presents multiresolution schemes for the efficient numerical solution of one-dimensional conservation laws with viscosity. The method, originally developed by A, Harten (Commun. Pure Appl. Math., to appear) for hyperbolic conservation laws, computes the cell average multiresolution representation of the solution which provides much information about the solution's regularity. As a consequence, the possibly expensive ENO (essentially nonoscillatory) reconstruction as well as numerous flux computations are performed only near discontinuities, and thereby the numerical solution procedure becomes considerably more efficient. The multiresolution scheme is also expected to ''follow'' possibly unsteady irregularities from one time step to the next. When viscosity is added, predicting the location of the irregularity becomes a problem of estimating the change in shock thickness. To this end, we derive shock width estimates for our 1D prototype equations, which, when combined with the stability restriction of the numerical scheme, provide a reliable mechanism for enlarging the original multiresolution stencil. The numerical experiments for scalar conservation laws indicate the feasibility of multiresolution schemes for the viscous case as well. (C) 1996 Academic Press, Inc.