On the use of AGE algorithm with a high accuracy Numerov type variable mesh discretization for 1D non-linear parabolic equations

In this paper, the use of N-AGE and Newton-N-AGE iterative methods on a variable mesh for the solution of one dimensional parabolic initial boundary value problems is considered. Using three spatial grid points, a two level implicit formula based on Numerov type discretization is discussed. The local truncation error of the method is of O(k²h_l^-1 +kh_l +h_l³)O({k^2h_l^{-1} +kh_l +h_l^3}), where h _l  > 0 and k > 0 are the step lengths in space and time directions, respectively. We use a special technique to handle singular parabolic equations. The advantage of using these algorithms is highlighted computationally.