论文标题
通过对广义KDV方程的隐式惩罚，一种新的保守性不连续的Galerkin方法
A new conservative discontinuous Galerkin method via implicit penalization for the generalized KdV equation
论文作者
论文摘要
我们设计，分析和实施了一种新的保守性不连续的盖尔金（DG）方法，以模拟对广义Korteweg-De Vries（KDV）方程的孤立波解决方案。我们方法的关键特征是在数值水平上的质量，能量和哈密顿量的保护，这些保存是通过所有KDV方程的精确溶液保守的。据我们所知，这是保存所有这三个数量的第一个DG方法，这对于孤立波的准确演变至关重要。为了获得所需的保护特性，我们的新思想是在数值通量中引入两个稳定参数，作为新的未知数，然后使我们能够在数值方案的制定中执行能量和哈密顿量的保护。我们证明了该方案的保护特性，这些属性通过数值测试证实了。通过隐式定义惩罚参数（传统上指定先验的惩罚参数）来实现保护属性的这种想法可以作为设计用于其他类型问题的物理保存数值方法的框架。
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.
