Splitting spectral element method for fractional reaction-diffusion equations

In this paper, we propose a second-order operator splitting spectral element method for solving fractional reaction-diffusion equations. In order to achieve a fast second-order scheme in time, we decompose the original equation into linear and nonlinear sub-equations, and combine a quarter-time nonlinear solver and a half-time linear solver followed by final quarter-time nonlinear solver. The spatial discretization is eigen-decomposition based on spectral element method. Since this method gives a full diagonal representation of the fractional operator and gets an exponential convergence in space. We have an accurate and efficient approach for solving spacial fractional reaction-diffusion equations. Some numerical experiments are carried out to demonstrate the accuracy and efficiency of this method. Finally, we apply the proposed method to investigate the effect of the fractional order in the fractional reaction-diffusion equations.

在本文中，我们提出了一种用于求解分数阶反应 - 扩散方程的二阶算子分裂谱元方法。为了在时间上实现快速的二阶格式，我们将原始方程分解为线性和非线性子方程，并结合四分之一时间的非线性求解器、二分之一时间的线性求解器以及最后的四分之一时间的非线性求解器。空间离散化是基于谱元方法的特征分解。由于该方法给出了分数阶算子的完全对角表示，并在空间上获得指数收敛，所以我们有一种精确且高效的方法来求解空间分数阶反应 - 扩散方程。我们进行了一些数值实验以证明该方法的准确性和高效性。最后，我们应用所提出的方法来研究分数阶在分数阶反应 - 扩散方程中的影响。