Journal of Scientific Computing ( IF 2.5 ) Pub Date : 2021-01-01 , DOI: 10.1007/s10915-020-01370-2 Chao Zhang , Yan Xu , Yinhua Xia
In this paper, we develop and analyze a series of conservative and dissipative local discontinuous Galerkin (LDG) methods for the dispersive system of Korteweg–de Vries (KdV) type equations. Based on a cardinal conservative quantity of this system, we design and discuss two different types of numerical fluxes, including the conservative and dissipative ones for the linear and nonlinear terms respectively. Thus, one conservative together with three dissipative LDG schemes for the KdV-type system are developed in our paper. The invariant preserving property for the conservative scheme and corresponding dissipative properties for the other three dissipative schemes are all presented and proven in this paper. The error estimates for two schemes are given, whose numerical fluxes for linear terms are chosen as the dissipative type. Assuming that the discontinuous piecewise polynomials of degree less than or equal to k are adopted, and conservative numerical fluxes are employed to discretize the nonlinear terms, we obtain a suboptimal a priori bound of order k; yet in the case of dissipative fluxes, we obtain a slightly better bound of order \(k+\frac{1}{2}\). Numerical experiments for this system in different circumstances are provided, including accuracy tests for two kinds of traveling waves, long-time simulations for solitary waves and interactions of multi-solitary waves, to illustrate the accuracy and capability of these schemes.
中文翻译:
KdV型方程的色散系统的局部不连续Galerkin方法
在本文中,我们开发和分析了一系列针对Korteweg-de Vries(KdV)型方程组的色散系统的保守和耗散局部不连续Galerkin(LDG)方法。基于该系统的基本保守量,我们设计和讨论了两种不同类型的数值通量,分别包括线性项和非线性项的保守性和耗散性。因此,本文针对KdV型系统开发了一种保守的方法以及三种耗散的LDG方案。本文给出并证明了保守方案的不变保性和其他三种耗散方案的相应耗散性。给出了两种方案的误差估计,将其线性项的数值通量选择为耗散类型。采用k,并采用保守的数值通量来离散非线性项,我们得到次优k阶的先验界;但是在耗散通量的情况下,我们获得阶数更好的界\(k + \ frac {1} {2} \)。提供了该系统在不同情况下的数值实验,包括两种行波的精度测试,孤波的长时间仿真以及多孤波的相互作用,以说明这些方案的准确性和能力。