An implicit high accuracy variable mesh scheme for 1-D non-linear singular parabolic partial differential equations

In this paper, we propose a new two-level implicit difference scheme of O(k2hl-1+khl+hl3) for the solution of non-linear parabolic equation εuxx = ϕ(x, t, u, ux, ut), 0 < x < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions prescribed, where k > 0, hl > 0 are mesh sizes in t- and x-coordinates respectively and ε > 0 is a small parameter. In addition, we also discuss a new explicit variable mesh difference scheme of O(khl+hl3) for the estimates of (∂u/∂x). In all cases, we require only three spatial variable grid points. The proposed schemes require less algebra and three evaluations of function ϕ. A special technique is required to solve singular parabolic equations. The proposed variable mesh scheme when applied to a linear diffusion equation is shown to be stable for all hl > 0 and k > 0. Computational results are provided to support our derived schemes and analysis.