摘要:
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-∂/∂x[a(x,y,t)∂u/∂x-∂/∂yo[nb(x,y,t)∂u/∂∂ycs]B=f本文研究两个着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),改进了已有结果。
Abstract:
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-∂/∂x[a(x,y,t)∂u/∂x-∂/∂yo[nb(x,y,t)∂u/∂ycs]B=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, L2 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 L2 norm and H1 semi-norm". In this paper, we try to improve these conclusions by H1 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(Δt2＋h2) in discrete H1 norm. This implies essential improvement of existing conclusions.
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