High accuracy difference schemes for the system of two space nonlinear parabolic differential equations with mixed derivatives and variable coefficients

In this article, two-level compact implicit difference methods of O(k2 + kh2 + h4) using 9-spatial grid points are proposed for the numerical solution of the system of two-dimensional nonlinear parabolic equations with variable coefficients subject to the Dirichlet boundary conditions, where k > 0 and h > 0 are step lengths in time and space directions, respectively. The proposed difference method for scalar equation is applied for the solution of the heat conduction equation in polar coordinates to obtain the two-level unconditionally stable ADI method of O(k2 + h4). The method having two variables also has been successfully applied on two-dimensional unsteady Navier-Stokesʹ model equations in polar coordinates. Some numerical examples are presented to demonstrate the accuracy of the implementation.