@Article{AAMM-1-5,
author = {Stefan, Berres and Raimund, Bürger and Kozakevicius, Alice},
title = {Numerical Approximation of Oscillatory Solutions of Hyperbolic-Elliptic Systems of Conservation Laws by Multiresolution Schemes},
journal = {Advances in Applied Mathematics and Mechanics},
year = {2009},
volume = {1},
number = {5},
pages = {581--614},
abstract = {<p style="text-align: justify;">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 $2 \times 2$ 
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.</p>},
issn = {2075-1354},
doi = {https://doi.org/10.4208/aamm.09-m0935},
url = {https://global-sci.com/article/73704/numerical-approximation-of-oscillatory-solutions-of-hyperbolic-elliptic-systems-of-conservation-laws-by-multiresolution-schemes}
}