Asymptotic Preserving Numerical Methods For Multiscale Transport And Waves

The multiscale problems and problems with interfaces are important since theyarise in many physical applications and attract a lot of physicians and mathemati-cians. One way to solve multiscale problems is to develop Asymptotic Preserving (AP)schemes. For PDEs with small parameters, a scheme is AP if it possesses the discreteanalogy of the continuous asymptotic limit as the small parameter goes to zero. APmethods are attractive since they work for all range of corresponding parameter andproblems involving di?erent scales. They avoid the interface conditions used in mostother multiscale methods to connect models of di?erent scales and supply a solverwhen it is di?cult to get such connection conditions.The thesis is focused on the AP schemes of the multiscale neutron transport equa-tion with interfaces and multiscale complex Ginzburg-Landau (CGL) equation, consid-ering the di?usion limit of the neutron transport equation with interfaces and the largetime and space scale limit of the CGL equation. The transport equation with interfaceshas a lot of important applications, ranging from high frequency wave in random mediato semiconductor simulation. For di?erent physical applications, there are two kindsof interfaces: the density continuous interface and the energy ?ux continuous interface.We derive the interface conditions of the di?usion limit for the density continuous caseand find two corresponding AP methods: exponential fitting method (EFM) and theimmersed interface method (IIM). EFM is not only AP, but second order convergentuniformly with respect to the mean free path in the whole domain. This is 1) the firstsharp uniform convergence result for linear transport equations in di?usive regime, aproblem that involves both transport and di?usive scales; and 2) the first uniform con-vergence valid up to the boundary even if the boundary layers exist. The CGL has along history in physics as a generic amplitude equation near the onset of instabilitiesthat lead to chaotic dynamics in ?uid mechanical systems. For this equation, we showthe AP property of the time splitting spectral method (TSSP) for the large time and space scale limit, while it is not AP for the nonlinear Shro¨dinger limit.Our numerical experiments show that all these AP schemes can capture the correctphysical behavior without resolving the small scale dynamics, even for problems wheresmall and large scales coexist, which saves a lot of computation time.