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

We present a new two-level compact implicit numerical scheme based on off-step discretization for the solution of the system of three space dimensional quasi-linear parabolic partial differential equations subject to suitable initial and boundary conditions. Further, we derive the estimates of first order space derivatives of the solution. The proposed methods are fourth order accurate in space and second order accurate in time, and involve 19-spatial grid points of a single compact cell. Also, we develop the alternating direction implicit (ADI) scheme for a linear parabolic equation with variable coefficients, which is shown to be unconditionally stable for the heat equation in polar coordinates. The methods so proposed do not require any modification when applied to the singular problems at the point of singularity unlike the numerical scheme proposed earlier in Mohanty and Jain (1994), and Mohanty (1997, 2003). The proposed methods are directly applicable to parabolic equations with singular coefficients. This is the main highlight of our work. The method successfully works for the Navier-Stokes equations of motion in polar coordinates. Many physical problems are solved to illustrate the accuracy of the proposed methods.