交替方向隐格式稳定性和收敛性的改进

Improvement on Stability and Convergence of A. D. I. Schemes

交替方向隐格式是数值求解高维抛物型方程的主要方法之一，考虑二维变系数抛物型方程ｕｔ－ｘａ（ｘ，ｙ，ｔ）ｕｘ－ｙｂ（ｘ，ｙ，ｔ）ｕｙ＝ｆ本文研究两个著名的交替方向隐式差分格式———Ｐ＿Ｒ格式和Ｄｏｕｇｌａｓ格式的稳定性和收敛性，对常系数情形（即函数ａ和ｂ均为常数），文献已证明了按离散Ｌ２范数的绝对稳定性和二阶收敛性，结论是完善的，但所用Ｆｏｕｒｉｅｒ分析方法不能推及一般变系数问题·文献采用了能量方法研究Ｐ＿Ｒ格式的稳定性和收敛性，但由于目的是Ｌ２估计以及使用了“Ｌ２范数与Ｈ１半范数等价”，所得到的Ｌ２稳定性和收敛性结论是很不完善的·本文采用Ｈ１能量估计方法，证明了格式按离散Ｈ１范数是稳定的，并且收敛阶为Ｏ（Δｔ２＋ｈ２）。

Alternating direction implicit(A. D. I. )schemes have been proved valuable in the approximation of the solutions of parabolic partial differential equations in multi_dimensional space. Consider equations in the form ut-xa(x,y,t)ux-yb(x,y,t)uy=f Two A. D. I. schemes, Peaceman_Rachford scheme and Douglas scheme will be studied. In the literature, stability and convergence have been analysed with Fourier Method, which cannot be extended beyond the model problem with constant coefficients. Additionally, L 2 energy method has been introduced to analyse the case of non_constant coefficients, however, the conclusions are too weak and incomplete because of the so_called “equiverlence between L 2 norm and H 1 semi_norm”. In this paper, we try to improve these conclusions by H 1 energy estimating method. The principal results are that both of the two A. D. I. schemes are absolutely stable and converge to the exact solution with error estimations O( Δ t 2+h 2) in discrete H 1 norm. This implies essential improvement of existing conclusions.