Adaptive sparse grids for hyperbolic conservation laws

We report on numerical experiments using adaptive sparse grid discretization techniques for the numerical solution of scalar hyperbolic conservation laws. Sparse grids are an efficient 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)\ logh \(d-1) points due to a truncated, tenser-product multi-scale basis representation. For the treatment of conservation laws two different approaches are taken: First an explicit time-stepping scheme based on central differences 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 refinement 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 refinement and local time-steps in a natural way.