@inproceedings{Griebel.Zumbusch:1999*1,
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},
ps = {http://wissrech.ins.uni-bonn.de/research/pub/zumbusch/hyp7.ps.gz 1},
pdf = {http://wissrech.ins.uni-bonn.de/research/pub/zumbusch/hyp7.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.}
}