A new spline in compression approximation for one space dimensional quasilinear 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 compression approximation for the numerical solution of one space dimensional quasi-linear parabolic partial differential equation on a uniform mesh. The derivation and the stability of the proposed method are discussed in details. We have extended the method to non-uniform mesh. Numerical results are given to illustrate the usefulness of the proposed method.