利用重心有理插值配点法求解一、二维对流扩散方程

On Barycentric Rational Interpolation Collocation Method to Solve 1D and 2D Convection Diffusion Equation

对流扩散是自然界中一种最为常见的物理现象,在气液固中均可发生,该方程已被广泛应用于飞行器设计、热磁辐射、天气预报、化工反应、生物斑点生长等重要领域.为了进一步提高该微分方程的逼近精度,可通过改进基底函数或者调整离散点分布来实现.借助于重心有理插值数值逼近稳定性好,离散矩阵具有稀疏性等优势,求解了一、二维对流扩散方程.将该数值方法与传统的FDM以及Meshfree等方法进行比较,得到的结论是:重心有理插值配点法在求解一、二维对流扩散问题上具有精度高、条件数小、收敛快等优点.从插值节点的分布效果上看,Chebyshev点比等距网格点更稳定,逼近精度略高,且能有效地抑制“龙格”现象的发生.最后,给出了相应的误差估计与收敛性分析,并使用软件画出了热流密度的分布云图,该图有利于分析对流扩散方程的数值解变化趋势问题.

Convection-diffusion problem is one of the most common physical phenomenon in nature,which can occur in gas,liquid,and solid.This equation has been widely used in aircraft design,thermal magnetic radiation,weather forecasting,chemical reactions,and biological spot growth and so on.Constructing a new basis function and adjusting the distribution of discrete points are common methods to improve the approximation accuracy of differential equations.The Barycentric Rational Interpolation method is employed to solve the one and two-dimensional convection-diffusion equation based on its higher precision and the sparse discrete matrix.By comparing the mentioned numerical method with the traditional FDM and Meshfree methods,the results show that the Barycentric Rational Interpolation collocation method has the advantages of high accuracy,small condition number,and fast convergence as solving the one and two-dimensional convection-diffusion equation.From the distribution effect of interpolation nodes,not only the stability and approximation accuracy of Chebyshev points are more superior than the equidistant grid points,but it can also effectively avoid“Runge phenomenon”.Finally,the relative error estimate and convergence analysis are given and the heat flux density cloud map is demonstrated,which is beneficial to analyze the change trend of the numerical solution of the convection-diffusion equation.