分数阶守恒Swift-Hohenberg方程的三种显隐Runge-Kutta方法

Three Explicit and Implicit Runge-Kutta Methods for Fractional Conservation Swift-Hohenberg Equation

摘要: 分数阶守恒Swift-Hohenberg (SH)方程是材料学中模拟凝固微观组织的基本模型。由于分数阶导算子及非局部守恒项的影响，许多求解经典整数阶SH方程行之有效的数值方法在解决此类问题时存在严重困难。本文针对分数阶守恒SH方程，研究其高效数值逼近算法。首先，在时间方向采用显隐Runge-Kutta方法，空间方向采用傅里叶谱方法，构造分数阶守恒SH方程的数值格式；其次，进一步给出所建立格式质量守恒的理论分析；最后，通过数值实验验证了格式的收敛阶和能量递减性，同时对长时间动力行为进行模拟，验证了算法的有效性。

Abstract: Fractional conservation Swift-Hohenberg (SH) equation is a basic model for simulating solidification microstructure in material science. Due to the influence of fractional derivative operators and nonlocal conservation terms, many effective numerical methods for solving classical integer order SH equations have severe difficulties in solving such problems. In this paper, an efficient numerical approximation algorithm for fractional conservative SH equation is studied. Firstly, the explicit and implicit Runge-Kutta method is used in the time direction and Fourier spectrum method is used in the space direction to construct the numerical schemes of fractional order conserved SH equation. Secondly, the theoretical analysis of mass conservation of these schemes is given. Finally, the convergence order and energy decrease of these schemes are verified by numerical experiments, and the validity of these algorithms are verified by long-term dynamic behavior simulation.