基于Chebyshev多项式的Galerkin谱方法在求解偏微分方程问题中的应用

Galerkin Spectral Method Based on Chebyshev Polynomials and Its Application on Numerical Solution of Partial Differential Equations

摘要: 基于Chebyshev多项式的Galerkin谱方法广泛应用于偏微分方程边值问题与初边值问题的计算 中，但详细介绍该方法的具体应用过程的文章较少。 本文通过求解具体例子(Helmholtz方程 边值问题、 含时一阶波动方程的初边值问题以及含时二阶线性热传导方程的初边值问题)来详 细介绍基于Chebyshev多项式的Galerkin谱方法的实现过程。 先假定方程的未知函数能够用基 于Chebyshev多项式展开式来逼近，然后将该未知函数的逼近展开式代入微分方程之中，再取方 程的弱形式并使其为零，进而得到未知函数展开式中的系数所满足的方程组，最终通过求解该方 程组得到未知函数的近似信息. 基于Chebyshev 多项式的Galerkin谱方法具有精度高、实现过程 简单等优点，本文通过算法实现过程及数值例子介绍了基于Chebyshev多项式的Galerkin 谱方 法的这些优点。

Abstract: Galerkin spectral method based on Chebyshev polynomials has been widely used to numerically solve the boundary value problem and initial boundary value problem of partial differential equation. However detailed introduction of the method and its application have been rarely seen in Chinese Journals. In this paper, we present the detailed implementation procedure of Chebyshev Galerkin spectral method by means of solving the boundary value problem of Helmholtz equation, initial boundary value problem of time-dependent Schrodinger equation and initial boundary value problem of wave equation, respectively. Our algorithm is built on: First we assume that the unknown function can be approximated by the expansion of Chebyshev polynomials; next we plug this expansion into the differential equation; then we use the weak formu- lation of the equation and make it zero, and obtain the discrete system which satisfied the coefficients of approximation expansion of unknown function; finally solving the discrete system gives us the approximated value of unknown function. Galerkin spec- tral method based on Chebyshev polynomials has the merit of high-order accuracy, and simple implement procedure. Our numerical algorithm and numerical examples have shown all of these merits of the Chebyshev spectral collocation method.