For Zoom Link please contact ddelreyfernandez@uwaterloo.ca
Speaker
Sigrun Ortleb, Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Kassel
Title
Numerical simulation of shallow water flows subject to wetting and drying by discontinuous Galerkin methods
Abstract
In coastal engineering and marine ecosystems, advancing and receding water fronts are an important aspect demanding a robust and efficient numerical simulation. Desired numerical properties include non-negativity of the water depth, mass conservation, well-balancedness with respect to lake at rest steady states, and avoidance of artificial pressure gradients.
In this talk, we focus on the 2D depth-averaged shallow water equations which are commonly used in coastal areas. These equations are based on the assumption of a small vertical length scale compared to large horizontal ones and a hydrostatic pressure distribution. The 2D shallow water equations will then be spatially discretized by the discontinuous Galerkin method on fixed unstructured grids. Well-balancedness with respect to lake at rest steady states requires modified volume terms and specific choices of numerical flux functions. While discretizing a skew-symmetric formulation of the momentum equation yields a well-balanced and entropy conservative scheme, preserving moving water equilibria requires a space discretization on staggered grids. We will also discuss non-conservative and pre-balanced formulations of the shallow water equations.
Regarding positivity preservation, a suitably small time step and certain limiting procedures usually guarantee the non-negativity of the water height in case of explicit time integrators. Nonetheless, in case of very small cells in nearly dry regions, explicit time integration might be inefficient. Hence, we consider implicit schemes which preserve the non-negativity of the water height for arbitrarily large time steps. From a theoretical point of view, the positivity preserving properties of the implicit Euler method are a key concept to the non-negativity of the water height. Furthermore, higher order unconditionally positive time integrators can be constructed based on the so-called Patankar approach which has gained renewed interest in recent works. Lastly, we will discuss how to avoid positivity preserving switches which might slow down the convergence of the nonlinear and linear solvers needed for implicit time integration.