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Fast IIF–WENO Method on Non-uniform Meshes for Nonlinear Space-Fractional Convection–Diffusion–Reaction Equations
Journal of Scientific Computing ( IF 2.5 ) Pub Date : 2021-08-25 , DOI: 10.1007/s10915-021-01622-9
Huan-Yan Jian 1, 2 , Ting-Zhu Huang 2 , Yong-Liang Zhao 2 , Alexander Ostermann 3 , Xian-Ming Gu 4
Affiliation  

In this article, our goal is to establish fast and efficient numerical methods for nonlinear space-fractional convection–diffusion–reaction (CDR) equations in the 1, 2, and 3 dimensions. For the spatial discretization of the CDR equations, the weighted essentially non-oscillatory (WENO) scheme is used to approximate the convection term, and the fractional centered difference formula is applied to deal with the diffusion term. As a result, a nonlinear system of ordinary differential equations (ODEs) is derived. Since implicit integration factor (IIF) methods are a class of time-stepping schemes with good robustness and stability, the second order IIF–WENO scheme on non-uniform meshes in Jiang and Zhang (J Comput Phys 253:368–388, 2013) is applied to solve the nonlinear ODEs system. In order to obtain an efficient implementation of the IIF–WENO scheme, we propose an adaptive restarting Krylov subspace method to compute the action of matrix exponentials arising in IIF–WENO. Numerical examples are presented to confirm the validity of the IIF–WENO scheme, and to verify that the proposed fast solution algorithm is extremely attractive in terms of computational complexity and memory storage.



中文翻译:

非线性空间分数对流-扩散-反应方程非均匀网格的快速 IIF-WENO 方法

在本文中,我们的目标是为 1、2 和 3 维的非线性空间分数对流扩散反应 (CDR) 方程建立快速有效的数值方法。对于CDR方程的空间离散化,采用加权基本非振荡(WENO)方案来近似对流项,并应用分数中心差分公式来处理扩散项。因此,推导出非线性常微分方程 (ODE) 系统。由于隐式积分因子 (IIF) 方法是一类具有良好鲁棒性和稳定性的时间步长方案,江和张的非均匀网格上的二阶 IIF-WENO 方案(J Comput Phys 253:368-388, 2013)用于求解非线性 ODE 系统。为了获得 IIF-WENO 方案的有效实现,我们提出了一种自适应重启 Krylov 子空间方法来计算 IIF-WENO 中出现的矩阵指数的作用。数值例子证实了 IIF-WENO 方案的有效性,并验证了所提出的快速求解算法在计算复杂性和内存存储方面极具吸引力。

更新日期:2021-08-26
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