一种基于Associated Hermite正交函数求解对流扩散方程的算法

提出了一种基于AH(Associated Hermite)正交基函数求解对流扩散方程的无条件稳定算法。该算法将方程的时间项通过Hermite多项式作为正交基函数进行展开,利用Galerkin方法消除时间变量项,从而导出有限维AH域隐式差分方程,突破了传统显式差分格式稳定性条件的限制,最后通过对AH域展开系数的求解得到该对流扩散方程的数值解。在数值算例中,将该算法与传统显示差分法和交替方向隐式差分法进行对比分析,数值计算结果表明,算法无条件稳定且其计算精度与时间步长无关,对于具有精细结构的对流换热问题,该算法具有明显的效率优势,且保持了较高的精度。

A new method for solving convection-diffusion equation using Associated Hermite orthogonal functions

In this work,an unconditionally stable method using the Associated Hermite (AH) orthogonal functions for solving the convection-diffusion equation is proposed.The time derivatives in the equation are expanded by the weighted Hermite functions.By introducing the Galerkin temporal testing procedure to the expanded equation,the time variable can be eliminated in the process of calculation.An implicit difference equation can then be obtained in AH domain under no convergent conditions.The numerical results of the equation can be obtained by solving the expanded coefficients in AH domain recursively.Two numerical examples were conducted to validate the accuracy and the efficiency of the proposed method by comparing to the conventional finite difference method and the alternating direction implicit (ADI) method.The numerical results have shown that the accuracy of this unconditionally stable method is independent of the time step size,and this proposed method has great advantage in efficiency in a computational domain with fine structure in convection-diffusion problems.Moreover,the agreement between the results obtained using the FD method and the proposed method is very good.