Improvement on stability and convergence of A. D. I. schemes

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

$$\frac{{\partial u}}{{\partial t}} - \frac{\partial }{{\partial x}}\left( {a(x,y,t)\frac{{\partial u}}{{\partial x}}} \right) - \frac{\partial }{{\partial y}}\left( {b(x,y,t)\frac{{\partial u}}{{\partial y}}} \right) = f$$

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 ”equivalence 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 0(Δt²+h²) in discrete H¹ norm. This implies essential improvement of existing conclusions.