DEDICATED TO THE MEMORY OF AMI HARTEN tion schemes have the potential to also simplify the grid generation process. In 1D the amount of time spent on This paper presents multiresolution schemes for the efficient numerical solution of one-dimensional conservation laws with viscos-grid generation is negligible (even for 2D Cartesian grids ity. The method, originally developed by A. Harten (Commun. Pure as shown by [2]); and, as is demonstrated by the short Appl. Math., to appear) for hyperbolic conservation laws, computes algorithms of [6], the multiresolution modules too are simthe cell average multiresolution representation of the solution which ple. For typical problems, the number of flux computations provides much information about the solution's regularity. As a can thus be reduced by as much as five or six times in both consequence, the possibly expensive ENO (essentially nonoscillatory) reconstruction as well as numerous flux computations are 1D [7, 1] and 2D [2].
performed only near discontinuities, and thereby the numerical so-
In this paper we modify the semi-discrete formulation lution procedure becomes considerably more efficient. The multiof the 1D multiresolution scheme [1] to solve the viscous resolution scheme is also expected to ''follow'' possibly unsteady model problem. We use the linear wave equation with irregularities from one time step to the next. When viscosity is viscosity (sometimes called the linearized Burgers' equaadded, predicting the location of the irregularity becomes a problem of estimating the change in shock thickness. To this end, we derive tion) and (viscous) Burgers' equation as our prototype shock width estimates for our 1D prototype equations, which, when linear and nonlinear equations, respectively. Adding viscombined with the stability restriction of the numerical scheme, cosity, in general, ''smoothes'' the solution, and in many provide a reliable mechanism for enlarging the original multiresolucases can even stabilize an otherwise unstable numerical tion stencil. The numerical experiments for scalar conservation laws scheme. Except for a time step limitation which can be indicate the feasibility of multiresolution schemes for the viscous case as well.