High order Methods for multidimensional systems of hyperbolic conservation laws using extrapolation

Description

Recently several first order methods have been developed for solving multidimensional nonlinear hyperbolic conservation laws, which are time accurate, which take multidimensional effects into account and which are not too costly. Consider for example LeVeque's Wave Propagation Method, the Fluctuation Splitting Method suggested by Roe, Deconinck and Struis or Fey's Method of Transport. However, in two or more space dimensions methods of high order are much more complex than first order methods.

The goal is to construct a simple method of variable order for hyperbolic systems of conservation laws via an adaptive approach, i.e. a method of high order in regions where the solution is smooth and of first order near shocks.

We compute on different grids using a first order method and obtain the desired accuracy by extrapolation. This technique already works well for scalar equations with variable coefficients in one space dimension in the interior of the domain. Problems occur in the treatment of boundary conditions. Special effort is needed to obtain conservation form between regions of different order.

As a typical example for a system of conservation laws the Euler equations in gas dynamics will be used. At a later stage an extension of the algorithm to Navier-Stokes will be done.

Contacts

Prof. Rolf Jeltsch and Hedwig Ulmer (ulmer@sam.math.ethz.ch).