求解一维线性双曲型方程的高精度紧致差分格式

A High-Order Compact Difference Scheme for Solving the One Dimensional Linear Hyperbolic Equation

【目的】双曲型方程是一类重要的偏微分方程,由于寻求问题本身的精确解比较困难,数值方法来求解此类方程有极具深远的意义和实际应用价值。【方法】首先对于一维的线性双曲型方程,在空间上采用Kreiss提出的四阶紧致差分公式进行逼近,时间上采用Taylor级数展开及截断误差修正的方法,推导出一个隐式的紧致差分格式。【结果】该格式在时间和空间上都有四阶精度,截断误差为O(τ4+h4)。【结论】采用Fourier方法分析了该格式的稳定性。数值实验证明提出的格式具有较好的稳定性和精确性。

[Purposes]Hyperbolic equations are an important class of partial differential equations.Because it is difficult to find the exact solution of the problem itself,use numerical methods for solving such equations.[Methods]Firstly,for the one dimensional linear hyperbolic equation,a high-order compact difference scheme is developed by using the fourth-order compact difference formula proposed by Kreiss in space direction and the Taylor series expansion and the truncation error correction method in time direction.[Findings]The truncation error of this method is O（τ4＋h4）.[Conclusions]Its stability criterion is determined by using Fourier analysis method.The numerical experiments are conducted and numerical results validate the stability and accuracy of the present scheme.