Single-cell discretization of O(kh2 + h4) for ∂u/∂n for three-space dimensional mildly quasi-linear parabolic equation

Abstract

In this article, using a single computational cell, we report some stable two-level explicit finite difference approximations of O(kh2 + h4) for ∂u/∂n for three-space dimensional quasi-linear parabolic equation, where h > 0 and k > 0 are mesh sizes in space and time directions, respectively. When grid lines are parallel to x-, y-, and z-coordinate axes, then ∂u/∂n at an internal grid point becomes ∂u/∂x, ∂u/∂y, and ∂u/∂z, respectively. The proposed methods are also applicable to the polar coordinates problems. The proposed methods have the simplicity in nature and use the same marching type of technique of solution. Stability analysis of a linear difference equation and computational efficiency of the methods are discussed. The results of numerical experiments are compared with exact solutions.