Hybrid multiscale methods for hyperbolic problems I. Hyperbolic relaxation problems

Hybrid Multiscale Methods for Hyperbolic and Kinetic Problems

In these notes we present some recent results on the development of hybrid methods for hyperbolic and kinetic equations with multiple scales. The main ingredients in the schemes are a suitable merging of particle methods in non stiff regimes with high resolution shock capturing techniques in stiff ones. The key aspect in the development of the algorithms is the choice of a suitable hybrid repre...

Asymptotic Preserving and Multiscale Methods for Kinetic and Hyperbolic Problems

Abstract. This presentation concerns the numerical approximation of a PDE system which models cell movements according to a chemoattractant concentration. The system under consideration turns out to couple a hyperbolic system with a diffusive equation. The solutions of such a model satisfy several properties to be preserved at the numerical level. Indeed, the solutions may contain vacuum, satis...

Spectral Methods for Hyperbolic Problems

We review spectral methods for the solution of hyperbolic problems. To keep the discussion concise, we focus on Fourier spectral methods and address key issues of accuracy, stability, and convergence of the numerical approximations. Polynomial methods are discussed when these lead to qualitatively different schemes as, for instance, when boundary conditions are required. The discussion includes...

A stable hybrid method for hyperbolic problems

A stable hybrid method for hyperbolic problems that combines the unstructured finite volume method with high-order finite difference methods has been developed. The coupling procedure is based on energy estimates and stability can be guaranteed. Numerical calculations verify that the hybrid method is efficient and accurate.

Recent Mathematical Results on Hyperbolic Relaxation Problems