Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation

  author = {M. Griebel and G. W. Zumbusch},
  title = {Adaptive Sparse Grids for Hyperbolic Conservation Laws},
  booktitle = {Hyperbolic Problems: Theory, Numerics, Applications. 7th
		  International Conference in Z\"{u}rich, February 1998},
  editor = {M. Fey and R. Jeltsch},
  volume = {1},
  pages = {411--422},
  series = {International Series of Numerical Mathematics 129},
  year = {1999},
  publisher = {Birkh\"{a}user},
  address = {Basel, Switzerland},
  ps = { 1},
  pdf = { 1},
  annote = {refereed series},
  abstract = {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 $h^{-1}|{\mathrm log}h|^{d-1}$ points due to a
		  truncated, tensor-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.}