A class of two- and three-level implicit methods of order two in time and four in space based on half-step discretization for two-dimensional fourth order quasi-linear parabolic equations

摘要

Novel compact schemes based on half-step discretization are developed to solve two dimensional fourth order quasi-linear parabolic equations. These discretizations use a single computational cell and the boundary conditions are incorporated in a natural way without the use of any fictitious points or discretization. The schemes have second order accuracy in time and fourth order accuracy in space. The proposed methods are directly applicable to problems constituting singular terms. We also present operator splitting method for the solution of linear parabolic equations which allows the use of the one-dimensional tri-diagonal solver multiple times. It is shown that the operator splitting method is unconditionally stable. The numerical schemes have been successfully applied to the two-dimensional Extended Fisher–Kolmogorov equation, Sivashinsky equation, Boussinesq equation and vibrations problem. The illustrative results confirm the theoretical order of magnitude and accuracy of the proposed methods.