An Efficient High-order Algorithm For Solving A Class Of Nonlinear Evolution Equation

In this thesis, we do some research of numerical approximate analysis for a class of two-dimensional and three-dimensional nonlinear evolution equation. In chapter 1, we introduce the preliminaries which will be used in this thesis. In chapter 2, basing on approximate factorization of difference operators, we apply the compact finite difference for the second derivative to derive a new finite difference scheme which is second order in time and fourth order in space for this kind of two-dimensional nonlinear evolution equation. Richardson extrapolation is applied to improve on accuracy in the temporal dimension. As a result, the new algorithm is easily extended to fourth-order accurate in both temporal and spatial dimensions. The new scheme is simple, and its computational cost is cheap. It only requires solutions of systems of tri-diagonal equations at each time level. It is shown through a discrete von Neumann analysis that the method is unconditionally stable. Numerical experiments are conducted to test the high accuracy and efficiency of the new algorithm. In chapter 3, motivated by the work of the second chapter, we develop an efficient higher order finite difference algorithm for solving 3D nonlinear evolution equation. The new algorithm is fourth-order accurate in both the temporal and spatial dimensions, and its computational procedure is simple. Numerical results are presented to indicate our new algorithm is robust, and have good numerical stability.