Non-polynomial Cubic Spline Methods For Solving One-dimensional Parabolic Equations

In this paper, based on the non-polynomial spline function, a family of difference schemes for solving one dimensional parabolic equation initial boundary value problems are constructed firstly. By choosing suitable parameters, many difference schemes may be derived from our methods, and we obtain a high accuracy three-level difference scheme, the accuracy of this scheme is fourth-order in time and space direction. Secondly, a new method is presented to deal with the Neumann boundary value problem, the accuracy of this method in space direction is improved to fourth-order. Finally, the numerical results show that our methods are very efficient.