Asymptotic preserving numerical schemes for transport and fluid equations

In certain asymptotic regimes many physical models can be accurately approximated by another model. However, the two models often have very different properties and require different strategies for numerical simulation. Asymptotic preserving (AP) schemes are schemes that can capture at the discrete level the transition from one model to the other, while only resolving time and space scales relevant to the regime at interest. One example of a model with such an asymptotic limit is kinetic models of particle transport through a background material. As the mean free path between particle collisions with the medium becomes small, over long time scales the predominant macroscopic behavior of the system is diffusive. However, when simulating multi-scale problems with regions of both high and low collisionality, such as multiple materials, a standard kinetic solver will require expensive numerical resolution of the mean free path in the diffusive regime, while the relevant scales of the underlying diffusive dynamics are independent of the mean free path. Another example is the low Mach number limit of the compressible Euler and Navier-Stokes equations. As the characteristic Mach number of the system becomes small solutions of the compressible equations can be approximated by solutions to the incompressible equations. However, in the low Mach number limit compressible solvers require resolution of the acoustic wave time scales of the system, despite the fact that these waves do not exist in incompressible regimes.;In this dissertation I investigated some problems arising in asymptotic preserving schemes for a linear model of diffusive transport. What we found was that with certain initial data, there is too little diffusion which causes unphysical oscillations to arise. We develop a program for correcting this diffusion that is still consistent with the asymptotic limit. We also develop an asymptotic perserving scheme for the isentropic Euler and Navier-Stokes equations that is suitable for both compressible and incompressible regimes. We present numerical results in compressible regimes that demonstrate its shock capturing ability and results in the incompressible regime that demonstrate that it captures the asymptotics without requiring resolution of the Mach number.