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

Abstract

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(k²h_l ^{− 1}+ kh_l + h_l ³) for the solution of non-linear parabolic equation u_xx = F(x,t,u,u_x, u_t) subject to appropriate initial and Dirichlet boundary conditions, where k > 0 and h_l > 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.

Keywords:

• Non-linear parabolic equation
• Variable mesh arithmetic average discretization
• A-AGE method
• Newton-A-AGE method
• Singular problems
• Burgers' equation

The author thanks the reviewers for their valuable suggestions, which substantially improved the standard of the paper.