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

Alternating direction implicit (A.D.I.) schemes have been proved valuable i n the approximation of the solutions of parabolic partial differential equa tions in multi-dimensional space. Consider equations in the form partial derivative u/partial derivative t - partial derivative/partial deri vative x(a(x,y,t) partial derivative u/partial derivative x) - partial deri vative/partial derivative y(b(x,y,t) partial derivative u partial derivativ e y) = f Two A.D.I. schemes, Peaceman-Rachford scheme and Douglas scheme will be stu died. In the literature, stability and convergence have been analysed with Fourier Method, which cannot be extended beyond the model problem with cons tant coefficients. Additionally, L-2 energy method has been introduced to a nalyse the case of non-constant coefficients, however, the conclusions are too weak and incomplete because of the so-called "equivalence between L-2 n orm 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 solutio n with error estimations O(Delta t(2) + h(2)) in discrete H-1 norm. This im plies essential improvement of existing conclusions.