Model-adaptive simulation of moment models with applications in shallow free-surface flows and rarefied gases
Many complex physical systems are described by high-dimensional partial differential equations (PDEs), whose numerical simulation can be computationally expensive. Lower-dimensional moment models offer a way to reduce this cost by retaining the essential physical behaviour while representing the system with a smaller number of variables. Their hierarchical structure makes them particularly suitable for adaptive simulations, in which the model resolution is varied in space and time according to the local physical complexity. This thesis develops a general framework for space- and time-adaptive simulation of moment models with a hierarchical structure. The framework includes methods for estimating the error introduced by reducing the model order and for coupling moment models of different orders within a single computational domain. It is applied to two classes of multiscale physical problems: shallow free-surface flows and rarefied gas dynamics. For shallow free-surface flows, a hyperbolic radially symmetric moment model is first developed, providing stable and increasingly accurate results with increasing model order. Next, an adaptive framework for hierarchical shallo water moment models is described. The adaptive framework is then used to dynamically select the model order in space and time, achieving accurate simulations while reducing computational cost by up to 60%. For rarefied gas dynamics, the adaptive framework is applied to a nonlinear hyperbolic kinetic moment model, with model selection based directly on the difference between successive model orders. The resulting simulations remain accurate compared with high-order models and benchmark solutions while reducing computational cost by up to 40%. Overall, the thesis demonstrates that adaptive moment models can provide an effective balance between physical detail and computational efficiency for multiscale simulations.