Well-Balanced and Invariant Domain Preserving Schemes for Dispersive Shallow Water Flows

Abstract

As urbanization encroaches more on flood prone regions and paved surfaces are ever expanding, more catastrophic flash floods occurring in urban environments are expected in the near future. These risks are compounded by global changes in the climate. Mathematics can help better predict and understand these situations through modeling and numerical simulations. The aim of this work is to discuss current mathematical and computational issues in modeling shallow water flows with applications in coastal hydraulics, large-scale oceanography and in-land flooding.

Our mathematical starting points are the systems of partial differential equations known as the (i) Saint-Venant shallow water equations and (ii) dispersive Serre–Green–Naghdi (SGN) equations. The goal of this work is to efficiently solve both mathematical models supplemented with external physical source terms for in-land flooding and large-scale coastal oceanography applications. In particular, the work focuses on introducing a novel technique for solving the Serre–Green–Naghdi equations. We introduce new analytical solutions of the SGN equations with topography that are used to verify the accuracy of numerical methods. Then, we propose a new relaxation technique for solving the SGN equations with topography effects that yields a hyperbolic formulation of the equations. This relaxation technique allows us to circumvent the dispersive time step restriction of the Serre Equations which is a major challenge when solving the equations. This method is then supplemented with a novel continuous finite element approximation that is second-order accurate in space, invariant domain preserving and well-balanced. The method is then verified with academic benchmarks and validated by comparison with laboratory experimental data.