Adaptive Multiresolution Schemes for Shock Computations

In this paper we present adaptive multiresolution schemes for the computation of discontinuous solutions of hyperbolic conservation laws. Starting with the given grid, we consider the grid-averages of the numerical solution for a hierarchy of nested grids which is obtained by diadic coarsening, and we compute its equivalent multiresolution representation. This representation of the numerical solution consists of the grid-averages of the coarsest grid and the set of errors in predicting the grid-averages of each level of resolution from the next coarser one; these errors depend on the size of the grid and the local regularity of the solution. At a jump-discontinuity they remain the size of the jump, independent of the level of resolution; this observation enables us to identify the location of discontinuities in the numerical solution. In a region of smoothness, once the numerical solution is resolved to our satisfaction at a certain locality of some grid, then the prediction errors there for this grid and all finer ones are small; this enables us to obtain data compression by setting to zero the terms of the multiresolution representation that fall below a specified tolerance. The numerical flux of the adaptive scheme is taken to be that of a standard centered scheme, unless it corresponds to an identified discontinuity, in which case it is taken to be the flux of an ENO scheme. We use the data compression of the numerical solution in order to reduce the number of numerical flux evaluations as follows: We start with the computation of the exact numerical fluxes at the few grid-points of the coarsest grid, and then we proceed through diadic refinement to the given grid. At each step of refinement we add values for the numerical flux at the new grid-points which are the centers of the coarser cells. Wherever the solution is locally well resolved (i.e., the corresponding prediction error is below the specified tolerance) the costly exact value of the numerical flux function is replaced by an accurate enough approximate value which is obtained by an inexpensive interpolation from the values of the coarser grid.