Effective Numerical Methods For Two Types Of High Order Partial Differential Equations

Numerical simulation of high order nonlinear partial differential equations isone of the major research problems in the field of scientific computing. As we allknow, many mathematical models of various scientific fields are mostly attributedto higher-order partial differential equations, Due to the complex of practicalproblems, lead to such equations with strongly nonlinear, small parameter,high order and many other difficulties.This article is mainly for two types of equations: Allen-Cahn equation andthe Cahn-Hilliard equation, these two equations are nonlinear higher-orderequations and the equation itself has a strong physical properties: massconservation, the total energy decreases with time, greatly reduce the accuracyof the simulation, the urgent need is to construct a new stable and efficientnumerical methods in order to better study higher-order partial differentialequations.This paper focuses on efficient and stable numerical solution of these twoequations, the main body of this paper are given as follows:First， we introduce two types of simple equations and the simplest form,the physical background in preface. The second part is for the Allen-Cahnequations, it constructs efficient and stable implicit-explicit numerical scheme,in order to get a more accurate numerical solutions, for spatial discretization, weuse a fourth-order compact scheme. And we use a linear implicit-explicit schemesto discrete time, and discussion the scheme stability; in order to have betterstability, we give the implicit-explicit Runge-Kutta scheme and emphasis provethe stability of the first-order and second-order scheme. The format of this articlecan prove superior to conventional numerical methods. Third part we give adifferential format for Cahn-Hilliard equation, the proof of the stability analysisand an adaptive time-stepping in the fourth section gives numerical experimentsto analyze two types of schemes for equations for verification.