A Comparison of the Explicit and Implicit Hybridizable Discontinuous Galerkin Methods for Nonlinear Shallow Water Equations

Abstract

An explicit implementation of the hybridizable discontinuous Galerkin (HDG) method for solving the nonlinear shallow water equations is presented. We follow the common construction of the implicit HDG for nonlinear conservation laws, and then explain the differences between the explicit formulation and the implicit version. For the implicit implementation, we use the approximate traces of the conserved variables ( $${\widehat{\varvec{q}}}$$ ) to express the numerical fluxes in each element. Next, we impose the conservation of the numerical fluxes via a global system of equations. Using the Newton–Raphson method, this global system can be solely expressed in terms of the increments of the approximate traces in each iteration. For the explicit method, having $${\varvec{q}}_h$$ at each time level, we first obtain $${\widehat{\varvec{q}}}_h$$ such that the conservation of the numerical flux is satisfied. This will result in a nonlinear system of equations which is local to each edge of the mesh skeleton. Having the solution ( $${\varvec{q}}_h$$ , $$\widehat{{\varvec{q}}}_h$$ ) for the previous time step, we use the Runge–Kutta time integration method to obtain $${\varvec{q}}_h$$ in the next time step. Hence, the introduced explicit technique is based on local operations over the faces and elements of the mesh. Using different numerical examples, we show the optimal convergence of the solution of the explicit and implicit approach in $$L^2$$ norm. Finally, through numerical experiments, we discuss the advantages of the implicit and explicit techniques from the computational cost point of view.