Improvement on stability and convergence of ADI schemes (Q1305392)

Two alternating direction implicit (ADI) schemes namely, the Peaceman-Rachford scheme and Douglas scheme for the two-dimensional quasilinear parabolic problem are studied. In the literature, stability and convergence have been analysed with the Fourier method, which can not be extended beyond the model problem with constant coefficients. Additionally, the \(L^{2}\) 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 ``equivalence between the \(L^{2}\) norm and the \(H^{1}\) semi-norm''. In this paper the author tries to improve these conclusions by the \(H^{1}\) energy estimating method. The principal results are that both of the two ADI schemes are absolutely stable and converge to the exact solution with error estimations \(O (\triangle t^{2} +h^{2}) \) in discrete \(H^{1}\) norm.