Robust numerical approximation for non-standard solutions of hyperbolic systems

Description

Non-standard solutions arise when the limit solution of a hyperbolic system depends on the underlying small scale effect like dissipation or dispersion. Standard numerical schemes add intrinsic numerical dissipation and fail to resolve the physically meaningful solution. This project consists of three sub-projects:

1. Numerical approximation of non-strictly hyperbolic systems (P. LeFloch and S. Mishra): The entropy solution of non-strictly hyperbolic system relies on the underlying diffusive-dispersive approximation. Further information in the form of kinetic relations must be provided to characterize the solutions. We investigate the existence of such kinetic relations by numerically approximating a model MHD system with Hall effect. Ongoing research aims to obtain numerical kinetic relations for the full MHD system.

2. Robust numerical schemes for non-conservative hyperbolic systems (M. Castro, U.S. Fjordholm, S. Mishra and C. Pares): Non-conservative hyperbolic systems arise in many contexts like multi-phase flow and multi-layer shallow water equations. The definition of entropy solutions requires the interpretation of products of distributions. State of the art schemes like path conservative schemes fail to converge to the physically relevant solution. We design a new class of entropy conservative path conservative (ECPC) schemes that have better convergence properties as the numerical diffusion operators can be modeled on the underlying physical diffusion operator. Ongoing work focuses on the design of arbitrarily high-order entropy stable path conservative DG schemes and application of these schemes to simulate realistic scenarios.

3. Numerical methods for the initial boundary value problem for systems of conservation laws (S. Mishra, L. V. Spinolo and M. Svaerd): We design entropy-stable numerical schemes for the initial boundary value problems. Recent work focuses on designing numerical diffusion operators that can approximate the physically meaningful solution of a boundary value problem. Ongoing research concentrates on the design of very high-order schemes for multi-dimensional BVPs.

Publications / Preprints or SAM-Reports

• Non-classical shocks and numerical kinetic relations for a model MHD system, P. G. LeFloch and S. Mishra, Acta Mathematica scientia, 29(6), 2009, 1684-1702.
• Accurate numerical discretizations of non-conservative hyperbolic systems, U.S. Fjordholm and S. Mishra, SAM report, 2010-25.
• Entropy stable schemes for initial-boundary-value conservation laws, M. Svärd and S. Mishra, SAM report, 2010-48.
  

Contacts

S. Mishra

U. Fjordholm