A new high accuracy two-level implicit off-step discretization for the system of two space dimensional quasi-linear parabolic partial differential equations

In this article, we propose a two-level compact implicit off-step discretization for the solution of the system of 2D quasi-linear parabolic partial differential equations subject to suitable initial and boundary conditions. New methods for the estimates of first order space derivatives of the solution are also derived. These methods are fourth order accurate in space and second order accurate in time, and involve only nine spatial grid points of a single compact cell. We further develop the alternating direction implicit (ADI) scheme for a general linear parabolic equation which is shown to be unconditionally stable for the heat equation in polar coordinates. The proposed methods are directly applicable to singular problems without the need of any special technique, which is the main advantage of this work. The method is effectively applied to the time dependent Navier-Stokes' model equations in polar coordinates. Numerical examples are provided to illustrate the accuracy of the methods.