An explicit hybridized discontinuous Galerkin method for Serre–Green–Naghdi wave model

Abstract

Simulation of coastal water waves requires a relevant mathematical model and a proper numerical technique which can be used in irregular geometries. In this paper, we study the numerical solution to a class of irrotational Green–Naghdi (GN) equation using the hybridized discontinuous Galerkin method (HDG). We first use Strang splitting to decompose this problem into a hyperbolic and a dispersive sub-problem. For the hyperbolic sub-problem, we use an explicit HDG with Lax–Friedrichs numerical fluxes. We also include different types of boundary conditions in our formulation. Similar to other HDG implementations, our global unknowns are the numerical traces, i.e. the unknowns on the mesh skeleton. This choice can significantly reduce the number of global degrees of freedom. Solving this part of the problem involves Newton iterations at each time step. However, unlike the implicit versions of HDG, the element equations are decoupled and we do not need to solve a large global system of equations. On the other hand, for the dispersive sub-problem we need to solve a global system of equations, implicitly. Since HDG, only involves unknowns on the mesh skeleton, it can be a very efficient alternative for this implicit solve step. By coupling these two solvers, we obtain the solution to the Green–Naghdi equation. Next, we use this technique to solve a set of verification and validation examples. In the first example we show the convergence properties of the numerical method. Next, we compare our results with a set of experimental data for the nonlinear water waves in different situations. Finally, we present a two dimensional simulation result for a solitary wave passing over a cosine bump. We observe close to optimal convergence rates and a good agreement between our numerical results and the experimental data.