对流方程的六阶中心差分格式

有限差分方法是微分方程数值解法中发展最早、理论最完善、应用最广泛的计算方法之一．利用待定系数法构造了对流方程的中心有限差分格式，利用Taylor级数展开推导出了该差分格式的修正偏微分方程（MPDE），采用数值余项效应分析方法从空间离散方面改进了该格式．利用高阶TVD Runge-Kutta方法从时间离散方面改进了该格式．利用Richardson外推方法在不增加计算复杂度的前提下改革了原格式．数值实验表明本文讨论的3种方法在差分格式改进和优化中的有效性．本文讨论的方法也可以用于其他偏微分方程有限差分方法的构造中．

Sixth-order central-difference scheme for convection equations

Finite difference method is one of the most important methods in finding numerical solutions of partial differential equation, which has been developed long time ago and has the most complete theory and many applications. In this paper undetermined coefficients method is used to get a central finite difference scheme for convection equation. Modified partial differential equation （MPDE） of the finite difference scheme is deduced by using Taylor series expansion. By doing remainder effect analysis the original scheme is reconstructed from space discretion. High order TVD Runge-kutta method is used to modify the scheme in time discretion. Richadson extrapolation method is used to reconstruct the new scheme without increasing the calculation complexity. Numerical experiments show the efficiency of the three methods in finite difference scheme＇e modification and optimization. The methods discussed can be used in finite difference scheme construction for other partial differential equations.