High order spline collocation methods for solving 1-D parabolic PDEs

In this paper, we describe the application of univariate cubic and quintic splines for solving one-dimensional parabolic equations. The approach involves Numerov discretization of space together with time integration based on both the cubic and quintic splines. It turns out that the resulting methods, which are semi-discrete, are examples of the method of lines (MOL). Moreover, the proposed methods present global solutions and shown to be unconditionally stable and having convergence order 0(h^4) + 0(k^4) and 0(h^4) + 0(k^6), where h and k are the spatial and temporal increments, for cubic and quintic splines, respectively. In addition, the spline methods may be regarded as continuous extensions, in time, of some discrete methods, in the sense that they provide global, in time, approximations which reproduce the values given by the discrete methods at grid points. Test examples will be provided, for the linear and nonlinear cases, to computationally illustrate the high accuracy and efficiency of the proposed methods compared to the extant methods and are adequate for long time interval problems.