﻿ Sobolev方程的一个紧致差分格式 A Compact Finite Difference Scheme for Sobolev Equations

Pure Mathematics
Vol.07 No.01(2017), Article ID:19461,9 pages
10.12677/PM.2017.71001

A Compact Finite Difference Scheme for Sobolev Equations

Xin Jing*, Luming Zhang

School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing Jiangsu

Received: Dec. 17th, 2016; accepted: Jan. 2nd, 2017; published: Jan. 5th, 2017

Copyright © 2017 by authors and Hans Publishers Inc.

ABSTRACT

A compact finite difference scheme is presented for Sobolev equations. It is proved by the discrete energy method that the compact scheme is unconditionally stable and convergent in norm, and the order of convergence is. The numerical experiment results show that the theory is accurate.

Keywords:Sobolev Equations, Compact Finite Difference Scheme, Convergence

Sobolev方程的一个紧致差分格式

1. 引言

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(2)

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Sobolev方程是一个重要的数学物理方程，被用来描述流体穿过岩石或土壤及不同介质中的流体运动，土壤中湿气的迁移等 [1] [2] [3] 。由于实际问题中Sobolev方程的源函数、初边值或计算域会很复杂，故有效的求解方法是求其数值解。

2. 差分格式的建立

2.1. 一些记号

2.2. 差分格式

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3. 数值解的先验估计

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。则(26)式有：

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4. 差分格式的收敛性与稳定性

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5. 数值算例

Table 1. Errors of numerical solution and convergence rate of example at different step sizes

Figure 1. The surface of exact solution and numerical solution when t = 1 (h = 1/20, τ = 1/400)

Figure 2. The surface of absolute error when t = 1 (h = 1/20, τ = 1/400)

A Compact Finite Difference Scheme for Sobolev Equations[J]. 理论数学, 2017, 07(01): 1-9. http://dx.doi.org/10.12677/PM.2017.71001

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