A new conservative discontinuous Galerkin method via implicit penalization for the generalized KdV equation

Abstract

We design, analyze, and implement a new conservative Discontinuous Galerkin (DG) method for the simulation of solitary wave solutions to the generalized Korteweg-de Vries (KdV) Equation. The key feature of our method is the conservation, at the numerical level, of the mass, energy and Hamiltonian that are conserved by exact solutions of all KdV equations. To our knowledge, this is the first DG method that conserves all these three quantities, a property critical for the accurate long-time evolution of solitary waves. To achieve the desired conservation properties, our novel idea is to introduce two stabilization parameters in the numerical fluxes as new unknowns which then allow us to enforce the conservation of energy and Hamiltonian in the formulation of the numerical scheme. We prove the conservation properties of the scheme which are corroborated by numerical tests. This idea of achieving conservation properties by implicitly defining penalization parameters, that are traditionally specified a priori, can serve as a framework for designing physics-preserving numerical methods for other types of problems.