Application of AGE Method to High Accuracy Variable Mesh Arithmetic Average Type Discretization for 1D Non-linear Parabolic Initial Boundary Value Problems

In this article, we discuss the application of arithmetic average discretization alternating group explicit (A-AGE) method to a new two-level implicit variable mesh formula of O(k2hl − 1+ khl + hl 3) for the solution of non-linear parabolic equation uxx = F(x,t,u,ux, ut) subject to appropriate initial and Dirichlet boundary conditions, where k > 0 and hl > 0 are the variable step lengths in time and space directions, respectively. The proposed technique is also useful to solve parabolic singular problems. Our discretization procedure requires only 3-spatial grid points. A-AGE and Newton-A-AGE algorithms are discussed in detail. Comparison of numerical results of these methods with the corresponding successive over relaxation (SOR) and Newton-SOR methods is also included.