Numerical Approximation of Oscillatory Solutions of Hyperbolic-Elliptic Systems of Conservation Laws by Multiresolution Schemes (Advances in Applied Mathematics and Mechanics, 2018)
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 theabsence of a core well-posedness theory for these equations is thesensitivity of their solutions to the structure of a parabolicregularization when attempting to single out an admissible solution by the vanishing viscosity approach. There is, however, theoreticaland numerical evidence for the appearance of solutions that exhibitpersistent oscillations, so-called oscillatory waves, which are (ingeneral, measure-valued) solutions that emerge from Riemann data orslightly 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 themultiresolution 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 porousmedia and a system modeling the separation of a heavy-buoyantbidisperse 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.