Spectral Methods For The Allen-Cahn Equation And Cahn-Hilliard Equation

Spectral method is a kind of numerical methods for solving differential equations, which is widely used in the fields of fluid dynamics, quantum mechanics, material science, ecology etc. To obtain numerical solutions of time-dependent partial differential equations(PDEs), we generally use spectral method in space and finite difference method in time. Because of the existence of nonlinear and stiff problems, most computations have been limited to second order accuracy in time. Although the spectral accuracy is achieved in space, low order of accuracy in time will also affect the total accuracy of the numerical solution.In order to improve the accuracy in time, we will introduce the exponential time differencing fourth-order Runge-Kutta method(ETDRK4) in detail. ETDRK4 method works well for some PDEs, it is fast and accurate, and it allows us to take larger time steps in real computation. At the same time, we will also introduce the semi-implicit method so as to compare with ETDRK4 methods under the same conditions.In Chapter 1, the brief summary of spectral method is given firstly, then we introduce the research advance of spectral method, semi-implicit method and ETDRK4 method. Additionally, we put forward the main content of the dissertation. Chapter 2 mainly reviews Chebyshev spectral method and Fourier spectral method, including the corresponding definitions, properties, computation of differentiation matrix. Then we introduce two types of time discretization schemes: the semi-implicit method and the ETDRK4 method. In Chapter 3 and 4, the methods discussed above are applied to numerical simulation of Allen-Cahn equation and Cahn-Hilliard equation in one dimension. We compare different time discretization schemes from the aspects of error, convergence order in time and CPU time. The effectiveness of the corresponding schemes is analyzed. Besides, aiming at taking larger time steps in the actual calculation,we introduce first-order stabilized semi-implicit methods for Allen-Cahn equation and Cahn-Hilliard equation. We also discuss the stability of first order semi-implicit schemes and the impact of the stability parameters on the accuracy of the solution. Finally, these two types of spectral methods are extended to solve two-dimensional equations. In the last chapter, the main content of the paper is summarized and prospected.