Well-posed boundary conditions and energy stable discontinuous Galerkin spectral element method for the linearized Serre equations

Abstract

We derive a class of well-posed boundary conditions for the linearized Serre equations in one spatial dimension using the energy method. The boundary conditions are formulated such that they are amenable to high order numerical methods. The resulting initial boundary value problem (IBVP) is energy stable, facilitating the design of robust and arbitrarily accurate numerical methods. An energy stable and conservative discontinuous Galerkin spectral element method with simple upwind numerical fluxes is proposed for solving the IBVP. For the numerical approximation, we derive discrete energy estimates by mimicking the continuous energy estimates and provide a priori error estimates in the energy norm. Detailed numerical examples are presented to verify the theoretical analysis and demonstrate convergence of numerical errors.