一种基于Legendre正交函数求解对流扩散方程的方法

A Numerical Method for solving the Convection-Diffusion Equation Based on Legendre Orthogonal Function

提出了一种基于Legendre正交函数求解对流扩散方程的无条件稳定方法.方法将对流扩散方程中的各项基于Legendre基函数进行展开,利用各阶基函数的正交性质和Galerkin方法消除方程中的时间微分项,形成可求解的系数矩阵方程,最后通过求解各阶展开系数可重构数值结果.为全面评价该方法,分别设计了具有精确解的一维方程和具有精细结构的二维问题等2个算例.计算结果表明:方法能够实现无条件稳定,且具有较高精度,同时在求解含有精细结构的对流扩散问题时具有明显的效率优势.

An unconditionally stable method for solving the convection-diffusion equation based on Legendre orthogonal function has been proposed in this paper.The equation can be expanded analytically based on the Legendre polynomials.By applying a Galerkin’s method and using the orthogonal property of Legendre polynomials,the time variable can be eliminated from computations,which results in an implicit equation.After solving the equation recursively one can obtain the numerical results by using the expanded coefficients.A one-dimensional equation with analytical solution and a two-dimensional problem with fine structure were conducted to validate the presented method.The numerical results have shown that the proposed method is accuracy and unconditionally stable,moreover,the method is efficient when it comes to a computational area with fine structure in convection-diffusion problems.