Adaptive Sparse Grids for Hyperbolic Conservation Laws

We report on numerical experiments using adaptive sparse grid dis-cretization techniques for the numerical solution of scalar hyperbolic conservation laws. Sparse grids are an eecient approximation method for functions. Compared to regular, uniform grids of a mesh parameter h contain h ?d points in d dimensions, sparse grids require only h ?1 jloghj d?1 points due to a truncated , tensor-product multi-scale basis representation. For the treatment of conservation laws two diierent approaches are taken: First an explicit time-stepping scheme based on central diierences is introduced. Sparse grids provide the representation of the solution at each time step and reduce the number of unknowns. Further reductions can be achieved with adaptive grid reenement and coarsening in space. Second, an upwind type sparse grid discretization in d + 1 dimensional space-time is constructed. The problem is discretized both in space and in time, storing the solution at all time steps at once, which would be too expensive with regular grids. In order to deal with local features of the solution, adaptivity in space-time is employed. This leads to local grid reenement and local time-steps in a natural way. 1. Sparse Grids Sparse grids were introduced for the solution of elliptic partial diierential equations , see 4] and references in 2]. They provide an eecient approximation method of smooth functions, especially in higher dimensions. So far, Galerkin methods 2] and nite diierence schemes 3, 9] for elliptic problems on sparse grids have been investigated. There are also attempts to solve parabolic problems 1] and Navier-Stokes equations 9]. The multi-dimensional approximation scheme of sparse grids can be constructed as a subspace of the tensor-product of one-dimensional spaces represented by a hierarchical multi-resolution scheme 5]. Consider piecewise linear interpolants on a d-dimensional unit hyper-cube. We start with the one-dimensional hierarchical basis 11]. The space of functions on the regular grid of dyadic level l and mesh parameter h = 2 ?l can be represented by the space of all tensor-products of one-dimensional basis functions of support larger than 2 ?l?1. The corresponding sparse grid space consists of all products of hierarchical basis functions with support larger than a d-dimensional volume of size 2 ?l?1 , see Figure 1. This is