Numerical Methods For Multiscale Hyperbolic And Kinetic Equations

Download or read book Numerical Methods for Multiscale Hyperbolic and Kinetic Equations written by and published by . This book was released on 2012 with total page 318 pages. Available in PDF, EPUB and Kindle. Book excerpt: Hyperbolic and kinetic equations often have parameters that vary considerably over the region. In certain asymptotic regimes where the parameter is very small, the standard hyperbolic or kinetic solvers break down because of the prohibitive computational cost. This thesis explores two efficient methods --- Domain Decomposition methods and Asymptotic Preserving (AP) methods for these problems. The first part aims at constructing a domain decomposition formulation for the Jin-Xin relaxation system with two-scale relaxations, which is a prototype for more general physical problems such as phase transitions, river flows, kinetic theories etc.. We propose the interface condition based on the sign of the characteristic speed at the interface. A rigorous analysis on the L2 error estimate is presented, based on the Laplace Tranform, for the linear case with an optimal convergence rate. For the nonlinear case, using standard compactness argument, we are able to prove the asymptotic convergence of the solution of the original relaxation system to the unique entropy weak solution of the domain decomposition system. The interface condition is derived rigorously by matched asymptotic analysis for a general flux with an extension to the case when a standing shock is sticking to the interface. The second part focuses on the development of AP methods for kinetic equations in the high field regime where both the collision and field effect dominate the evolution. The stiff force term poses extra numerical challenges as apposed to the stiff collision term which has been well-studied in the hydrodynamic regime. We first consider the Vlasov-Poisson-Fokker-Planck system used in electrostatic plasma and astrophysics. The AP scheme is constructed based on the combination of two stiff terms so as to use the symmetric discretization. The semiconductor Boltzmann equation is considered next. By penalizing the collision term by a classical BGK operator and treating the force term implicitly, we are able to overcome the exceptional difficulty that no specific expression of the local equilibrium is available. The distribution function is still shown to converge to the high field limit, which guarantees the capturing of the asymptotics without numerically resolving the small parameter.