Model-adaptive simulation of moment models with applications in shallow free-surface flows and rarefied gases

Simulating high-dimensional partial differential equations requires heavy computing power. Hierarchical moment models ease this burden by retaining core physical behavior with fewer variables, allowing model resolution to adapt across space and time. This thesis develops a framework for adaptive moment simulations, featuring methods to estimate reduction errors and couple differing model orders within a single domain. The strategy was applied to shallow free-surface flows and rarefied gas dynamics. In shallow flows, a stable hyperbolic radially symmetric model was created, and adaptive order selection cut computational expense by up to 60% while maintaining accuracy. For rarefied gases, adapting a nonlinear hyperbolic kinetic model using differences between consecutive orders reduced computation by up to 40% while matching benchmark results.