求解抛物型方程的两层高精度差分格式

Two⁃Layer High⁃Precision Difference Scheme for Solving Parabolic Equations

抛物型偏微分方程在工程技术与自然科学领域中扮演着重要作用,特别是在渗流、热传导、扩散等领域。对抛物型方程进行数值解法研究,在网格剖分的基础上,先给出一个含参数的差分格式,利用泰勒级数展开法和待定系数法使该差分格式的截断误差达到O (τ~3+h~5),通过方程组确定参数,得到一个两层高精度差分格式;然后用Fourier分析法解出在此精度下达到稳定的条件,即r≤19+√1141/60;最后通过数值算例将此差分格式数值解与精确解进行了比较,验证了新方法是可行的和有效的。

Parabolic partial differential equations play an important role in the fields of engineering technology and natural science,especially in the fields of seepage,heat conduction and diffusion.This paper mainly studies the numerical solution of parabolic equations.On the basis of mesh segmentation.First,a differential format with parameters is given.The Taylor series expansion method and the undetermined coefficient method are used to make the truncation error of the difference format reach O(τ3+h5).The parameters are determined by the equations to obtain a two⁃layer high⁃precision difference format.Then the Fourier analysis method is used.Solve the condition that achieves stability under this precision,that is r≤19+√1141/60.The numerical solution is compared with the exact solution by numerical examples,which verifies that the new method is feasible and effective.