Multiresolution schemes for conservation laws |
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Authors: | Wolfgang Dahmen Birgit Gottschlich–Müller Siegfried Müller |
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Institution: | (1) Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany; e-mail: {dahmen,birgit,mueller}@igpm.rwth-aachen.de , DE |
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Abstract: | Summary. In recent years a variety of high–order schemes for the numerical solution of conservation laws has been developed. In general,
these numerical methods involve expensive flux evaluations in order to resolve discontinuities accurately. But in large parts
of the flow domain the solution is smooth. Hence in these regions an unexpensive finite difference scheme suffices. In order
to reduce the number of expensive flux evaluations we employ a multiresolution strategy which is similar in spirit to an approach
that has been proposed by A. Harten several years ago. Concrete ingredients of this methodology have been described so far
essentially for problems in a single space dimension. In order to realize such concepts for problems with several spatial
dimensions and boundary fitted meshes essential deviations from previous investigations appear to be necessary though. This
concerns handling the more complex interrelations of fluxes across cell interfaces, the derivation of appropriate evolution
equations for multiscale representations of cell averages, stability and convergence, quantifying the compression effects
by suitable adapted multiscale transformations and last but not least laying grounds for ultimately avoiding the storage of
data corresponding to a full global mesh for the highest level of resolution. The objective of this paper is to develop such
ingredients for any spatial dimension and block structured meshes obtained as parametric images of Cartesian grids. We conclude
with some numerical results for the two–dimensional Euler equations modeling hypersonic flow around a blunt body.
Received June 24, 1998 / Revised version received February 21, 2000 / Published online November 8, 2000 |
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Keywords: | Mathematics Subject Classification (1991): 65M12 65M55 42C15 47A20 76Axx 35L65 |
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