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We design, analyze, and implement a new conservative Discontinuous Galerkin(DG) method for the simulation of solitary wave solutions to the generalizedKorteweg-de Vries (KdV) Equation. The key feature of our method is theconservation, at the numerical level, of the mass, energy and Hamiltonian thatare conserved by exact solutions of all KdV equations. To our knowledge, thisis the first DG method that conserves all these three quantities, a propertycritical for the accurate long-time evolution of solitary waves. To achieve thedesired conservation properties, our novel idea is to introduce twostabilization parameters in the numerical fluxes as new unknowns which thenallow us to enforce the conservation of energy and Hamiltonian in theformulation of the numerical scheme. We prove the conservation properties ofthe scheme which are corroborated by numerical tests. This idea of achievingconservation properties by implicitly defining penalization parameters, thatare traditionally specified a priori, can serve as a framework for designingphysics-preserving numerical methods for other types of problems.

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