A new algorithm based on spline in tension approximation for 1D quasi-linear parabolic equations on a variable mesh

In this paper, we propose a new two-level implicit method of order two in time and four in space directions, based on spline in tension approximation for the numerical solution of one space dimensional quasi-linear parabolic partial differential equation on a uniform mesh. We have discussed the derivation of the proposed method in detail and have also discussed the stability analysis for a model problem. We have extended the method to non-uniform mesh. Numerical results are given to illustrate the usefulness of the proposed methods.