Numerical Approximation of Oscillatory Solutions of Hyperbolic-Elliptic Systems of Conservation Laws by Multiresolution Schemes
Numerical Approximation of Oscillatory Solutions of Hyperbolic-Elliptic Systems of Conservation Laws by Multiresolution Schemes
Authors
Berres, Stefan
Burger, Raimund
Kozakevicius, Alice
Burger, Raimund
Kozakevicius, Alice
Authors
Date
2012-03-07
Datos de publicación:
10.4208/aamm.09-m0935
Keywords
Ingeniería matemática
Collections
Abstract
The generic structure of solutions of initial value problems of hyperbolic-
elliptic systems, also called mixed systems, of conservation laws is not yet
fully understood. One reason for the absence of a core well-posedness theory for
these equations is the sensitivity of their solutions to the structure of a parabolic
regularization when attempting to single out an admissible solution by the vanishing
viscosity approach. There is, however, theoretical and numerical evidence for
the appearance of solutions that exhibit persistent oscillations, so-called oscillatory
waves, which are (in general, measure-valued) solutions that emerge from Riemann
data or slightly perturbed constant data chosen from the interior of the elliptic region.
To capture these solutions, usually a fine computational grid is required. In
this work, a version of the multiresolution method applied to a WENO scheme
for systems of conservation laws is proposed as a simulation tool for the efficient
computation of solutions of oscillatory wave type. The hyperbolic-elliptic 2×2 systems
of conservation laws considered are a prototype system for three-phase flow
in porous media and a system modeling the separation of a heavy-buoyant bidisperse
suspension. In the latter case, varying one scalar parameter produces elliptic
regions of different shapes and numbers of points of tangency with the borders of
the phase space, giving rise to different kinds of oscillation waves.