ASYMPTOTIC PRESERVING NUMERICAL SCHEMES FOR TRANSPORT AND FLUID EQUATIONS By

iAbstract 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 syste