A new stable variable mesh method for 1-D non-linear parabolic partial differential equations

We propose a new stable variable mesh implicit difference method for the solution of non-linear parabolic equation u_xx = φ{symbol}(x, t, u, u_x, u_t), 0 < x < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions prescribed. We require only (3 + 3)-spatial grid points and two evaluations of the function φ{symbol}. The proposed method is directly applicable to solve parabolic equation having a singularity at x = 0. The proposed method when applied to a linear diffusion equation is shown to be unconditionally stable. The numerical tests are performed to demonstrate the convergence of the proposed new method.