Model-Adaptive Simulation of Hierarchical Shallow Water Moment Equations in One Dimension

Abstract: Shallow free surface flows are often characterized by both subdomains that require high modeling complexity and subdomains that can be sufficiently accurately modeled with low modeling complexity. Moreover, these subdomains may change in time as the water flows through the domain. This motivates the need for space and time adaptivity in the simulation of shallow free surface flows. In this paper, we develop the first adaptive simulations using the recently developed Shallow Water Moment Equations, which are an extension of the standard Shallow Water Equations that allow for vertically changing velocity profiles by including additional variables and equations. The model-specific modeling complexity of a shallow water moment model is determined by its order. The higher the order of the model, the more variables and equations are included in the model. Shallow water moment models are ideally suited for adaptivity because they are hierarchical such that low-order models and high-order models share the same structure. To enable adaptive simulations, we propose two approaches for the coupling of the varying-order shallow water moment equations at their boundary interfaces. The first approach dynamically updates padded state variables but cannot be written in conservative form, while the second approach uses fixed padded state variable values of zero and reduces to conservative form for conservative moment equations. The switching procedure between high-order models and low-order models is based on a new set of model error estimators, originating from a decomposition of the high-order models. Numerical results of the collision of a dam-break wave with a smooth wave yield accurate results, while achieving speedups up to 60 percent compared to a non-adaptive model with fixed modeling complexity.