数值计算与计算机应用 2008, 29(2 ) 146-152 DOI:     ISSN: 1000-3266 CN: 11-2124/TP

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一维抛物型方程;;子域精细积分隐格式;;三次样条函数;;稳定性
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PubMed

一维抛物型方程的样条子域精细积分(SSPI)隐格式

刘利斌,刘焕文

广西民族大学数学与计算机科学学院;广西民族大学数学与计算机科学学院 南宁 530006;南宁 530006

摘要

对一维抛物型方程初边值问题的求解,以往已经有一些数值解法,它们或者无条件稳定但精度不高,或者精度高但仅为条件稳定,且稳定性条件严格.另外,以往的差分格式在处理第二、第三类边界条件问题时,对带导数边界条件都是进行简单的差分逼近,影响了数值解的精度.因此构造一个无条件稳定且对各类边值问题都具有良好精度的数值方法具有重要意义.为此,基于子域精细积分思想,结合三次样条函数,提出了求解一维抛物型方程初边值问题含参数的样条子域精细积分格式.该格式为绝对稳定且精度很高.由于三次样条函数的采用,避免了通常有限差分法中处理带导数边界条件时产生的逼近误差,大大提高了求解第二、三类边界条件问题时的精度.

关键词 一维抛物型方程;;子域精细积分隐格式;;三次样条函数;;稳定性  

A SPLINE SUB-DOMAIN PRECISE INTEGRATION IMPLICIT SCHEME FOR SOLVING ONE-DIMENSIONAL PARABOLIC EQUATIONS

Liu Libin Liu Huanwen (Faculty of Mathematics and Computer Science,Guangxi University for Nationalities,Nanning 530006,China)

Abstract:

There are some numerical methods to solve the one-dimensional initial-boundary value problem of parabolic equations.They are either unconditionally stable but not very ac- curate,either accurate but conditionally stable.Especially,when problems subject to the second and the third boundary conditions are considered,the approximation by using the simple difference scheme only to the bounday conditions with derivatives leads to the loss of accuracy.Hence,it is significant to construct an a numerical method which is not only unconditionally stable but also accurate to deal with various boundaey conditions.Based on the sub-domain precise integration method and the cubic spline function,a new method, called the spline sub-domain precise integration (SSPI) scheme containing a parameter for the one-dimensional initial-boundary value problem of parabolic equations is presented.Be- cause of the application of the cubic spline function,the error arosed in classical difference schemes to approximate boundary conditions with derevitives is avoided and the numerical accuracy to solve the problem with the second and the third boundary conditions is greatly improved.

Keywords: one-dimensional parabolic type equation;;sub-domain precise integration implicit scheme;;cubic spline function;;stability  
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