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.