﻿ 一维Sine-Gordon方程四阶紧致有限体积方法 A Fourth-Order Compact Finite Volume Scheme for 1D Sine-Gordon Equations

Vol.04 No.03(2015), Article ID:15897,9 pages
10.12677/AAM.2015.43032

A Fourth-Order Compact Finite Volume Scheme for 1D Sine-Gordon Equations

Angran Liu, Wei Gao, Hong Li

School of Mathematical Sciences, Inner Mongolia University, Hohhot Inner Mongolia

Received: Jul. 29th, 2015; accepted: Aug. 11th, 2015; published: Aug. 18th, 2015

ABSTRACT

In this work, we propose a compact finite volume method for solving the one-dimensional nonlinear sine-Gordon equation. The third-order SSP Runge-Kutta (RK) scheme is used for temporal disretization. Numerical experiments show that the present scheme is an efficient algorithm for solving the one-dimensional Sine-Gordon equation.

Keywords:Sine-Gordon, Compact Method, Finite Volume Method, Runge-Kutta Method

1. 引言

(1)

(2)

(3)

(4)

(5)

(6)

(7)

2. 紧致有限体积方法

(8)

，称在第个单元上的单元平均值，那么(8)式可以简记为

(9)

(10)

(11)

,

(12)

(13)

(14)

, ,

(15)

(16)

(17)

,

(18)

3. Runge-Kutta时间离散

(19)

R，P分别定义为空间上的线性和非线性的微分算符。时间格式的选取依赖于所要的精度阶，要考虑储存、计算、稳定性。尤其是当R，P是非线性的时候，更没有依据说明全离散后的格式是稳定的，这时就更需要强稳定时间格式，常见的高阶稳定离散可见[16] 。

2阶方程对应初值问题的一般形式为

(20)

(20)式对应的Runge-Kutta格式为

(21)

4. 数值算例

(22)

(23)

Table 1. Errors and orders for 4-order CFVM

Figure 1. Curve: numerical and exact solution of the first experiment numerical (square); exact (solid line)

Table 2. Errors and orders for 4-order CFVM

Figure 2. Curve: numerical and exact solution of the second experiment numerical (square); exact (solid line)

A Fourth-Order Compact Finite Volume Scheme for 1D Sine-Gordon Equations[J]. 应用数学进展, 2015, 04(03): 262-270. http://dx.doi.org/10.12677/AAM.2015.43032

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