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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2015, Vol. 32 ›› Issue (6): 876-882.doi: 10.3969/j.issn.1005-3085.2015.06.008

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KdV方程的一个紧致差分格式

赵修成,   黄浪扬   

  1. 华侨大学数学科学学院，福建 泉州 362021
  • 收稿日期:2014-04-28 接受日期:2014-10-28 出版日期:2015-12-15 发布日期:2016-02-15
  • 基金资助:
    国家自然科学基金 (11271171)；福建省自然科学基金 (2011J01010).

A Compact Difference Scheme for the KdV Equation

ZHAO Xiu-cheng,   HUANG Lang-yang   

  1. School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021
  • Received:2014-04-28 Accepted:2014-10-28 Online:2015-12-15 Published:2016-02-15
  • Supported by:
    The National Natural Science Foundation of China (11271171); the Natural Science Foundation of Fujian Province (2011J01010).

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摘要： 本文基于经典的有限差分方法，讨论了满足周期边界条件的KdV方程的高精度差分格式的构造问题．通过引入中间函数及紧致方法对空间区域进行离散，提出了KdV方程的一个两层隐式紧致差分格式．利用泰勒展开法得出，该格式在时间方向具有二阶精度，但在空间方向可达到六阶精度．采用线性稳定性分析法证明了该格式是稳定的．数值结果表明：本文所提出的紧致差分格式是有效的，在空间方向拥有较高的精度，还能够很好地保持离散动量和能量守恒性质．

关键词: KdV方程, 紧致差分格式, 截断误差, 稳定性分析, 数值例子

Abstract:

Based on the classical finite difference method, the paper discusses the construction of a high accuracy difference scheme for the KdV equation with periodic boundary conditions. By introducing an intermediate function and a compact method to discretize the space area, a two-layer implicit compact difference scheme for the KdV equation is proposed. Using the Taylor expansion method, we show that the proposed scheme has second order accuracy in time direction, but can reach sixth order accuracy in spatial direction. The linear stability analysis method proves the scheme is stable. Numerical results show that the compact difference scheme proposed in this paper is effective, it has high accuracy in the spatial direction, and can also keep the conservations of momentum and energy well.

Key words: KdV equation, compact difference scheme, truncation error, stability analysis, numerical example

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