Multiscale Hyperbolic Equations: Numerical Approximation and Applications

Abstract

Various problems arising throughout engineering and applied sciences involve multiscale in space or time. Classical examples include the transport equations, such as the neutron transport and the radiative transfer, gas dynamics far from thermo or chemical equilibrium, fluid flows at different Reynolds number. Moreover, in many physical applications, the scaling parameter, i.e., the mean free path in kinetic theory, may differ in several order of magnitude from the rarefied regime to the hydrodynamic (or diffusive) regime within the same problem. In this work we are interested in numerical techniques for solving kinetic equations in the diffusive regimes (although the approach considered here is applicable to many physical problems of greater complexity). We will illustrate these basic techniques by means of a few simple models when the limit state is described by a general reaction-diffusion system. Some applications are presented including porous media equation, Fisher’s equation and ion diffusion.KeywordsHyperbolic SystemDiffusive RegimePorous Media EquationRelaxation TermMacroscopic EquationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.