二维抛物方程基于降维格式的一种差分谱逼近

A difference spectral approximation based on the dimension reduction scheme for two-dimensional parabolic equations

针对圆域上的二阶抛物问题,提出了基于高阶多项式逼近的一种有效的数值方法。该方法的主要思想是利用极坐标变换及Fourier基函数展开,将原问题分解为一系列解耦的一维二阶抛物问题。然后,对每个一维二阶抛物问题,建立了一种弱形式及其离散格式,并从理论上证明了该格式的稳定性,弱解和逼近解的存在唯一性以及它们之间的误差估计。最后,给出了一些数值算例,数值结果表明了算法的稳定性和收敛性。

For the second-order parabolic problem in a circular domain,we propose in this paper an effective numerical method based on high-order polynomial approximation.The main idea of this method is to use polar coordinate transformation and Fourier basis function expansion to decompose the original problem into a series of decoupled one-dimensional second-order parabolic problems.Then,for each one-dimensional second-order parabolic problem,a weak form and its discrete scheme is established,and theoretically prove the stability of the schemes,the existence and uniqueness of the weak solution and the approximate solution,as well as the error estimate between them.Finally,some numerical examples is presented,and the numerical results show the stability and convergence of our algorithm.