Applied Numerical Mathematics ( IF 2.8 ) Pub Date : 2021-07-13 , DOI: 10.1016/j.apnum.2021.07.003 Xin Huang 1 , Hai-Wei Sun 1
In this paper, we develop a fast numerical method for solving the time-dependent Riesz space fractional diffusion equations with a nonlinear source term in the convex domain. An implicit finite difference method is employed to discretize the Riesz space fractional diffusion equations with a penalty term in a rectangular region by the volume-penalization approach. The stability and the convergence of the proposed method are studied. As the coefficient matrix is with the Toeplitz-like structure, the generalized minimum residual method with a preconditioner based on the sine transform is exploited to solve the discretized linear system, where the preconditioner is constructed in view of the combination of two approximate inverse τ matrices, which can be diagonalized by the sine transform. The spectrum of the preconditioned matrix is also investigated. Numerical experiments are carried out to demonstrate the efficiency of the proposed method.
中文翻译:
基于正弦变换的凸域二维半线性Riesz空间分数阶扩散方程的预处理器
在本文中,我们开发了一种快速数值方法,用于求解具有凸域非线性源项的瞬态 Riesz 空间分数扩散方程。采用隐式有限差分方法,通过体积惩罚方法在矩形区域内离散带有惩罚项的 Riesz 空间分数扩散方程。研究了所提出方法的稳定性和收敛性。由于系数矩阵具有类 Toeplitz 结构,利用基于正弦变换的带预处理器的广义最小残差法求解离散化线性系统,其中预处理器是针对两个近似逆τ的组合构造的矩阵,可以通过正弦变换对角化。还研究了预处理基质的光谱。进行数值实验以证明所提出方法的有效性。