Numerical correction of finite difference solution for two-dimensional space-fractional diffusion equations with boundary singularity

2020 ◽  
Author(s):  
Zhaopeng Hao ◽  
Wanrong Cao ◽  
Shengyue Li
Keyword(s):  
Finite Difference ◽  
Dimensional Space ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Two Dimensional ◽  
Boundary Singularity ◽  
Fractional Diffusion Equations ◽  
Finite Difference Solution ◽  
Difference Solution

Preconditioned Iterative Methods for Two-Dimensional Space-Fractional Diffusion Equations

2015 ◽  
Vol 18 (2) ◽  
pp. 469-488 ◽  
Author(s):  
Xiao-Qing Jin ◽  
Fu-Rong Lin ◽  
Zhi Zhao
Keyword(s):  
Linear Systems ◽  
Iterative Methods ◽  
Dimensional Space ◽  
Fractional Diffusion ◽  
Second Order ◽  
Diffusion Equations ◽  
Two Dimensional ◽  
Fractional Diffusion Equations ◽  
Preconditioned Iterative ◽  
Preconditioned Iterative Methods

AbstractIn this paper, preconditioned iterative methods for solving two-dimensional space-fractional diffusion equations are considered. The fractional diffusion equation is discretized by a second-order finite difference scheme, namely, the Crank-Nicolson weighted and shifted Grünwald difference (CN-WSGD) scheme proposed in [W. Tian, H. Zhou and W. Deng, A class of second order difference approximation for solving space fractional diffusion equations, Math. Comp., 84 (2015) 1703-1727]. For the discretized linear systems, we first propose preconditioned iterative methods to solve them. Then we apply the D’Yakonov ADI scheme to split the linear systems and solve the obtained splitting systems by iterative methods. Two preconditioned iterative methods, the preconditioned generalized minimal residual (preconditioned GMRES) method and the preconditioned conjugate gradient normal residual (preconditioned CGNR) method, are proposed to solve relevant linear systems. By fully exploiting the structure of the coefficient matrix, we design two special kinds of preconditioners, which are easily constructed and are able to accelerate convergence of iterative solvers. Numerical results show the efficiency of our preconditioners.

A Low-Dimensional Compact Finite Difference Method on Graded Meshes for Time-Fractional Diffusion Equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Rezvan Ghaffari ◽  
Farideh Ghoreishi
Keyword(s):  
Finite Difference ◽  
Dimensional Space ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Computational Time ◽  
Initial Time ◽  
Compact Finite Difference ◽  
Fractional Diffusion Equations ◽  
Graded Meshes ◽  
Cfd Method

Abstract In this paper, we propose an improvement of the classical compact finite difference (CFD) method by using a proper orthogonal decomposition (POD) technique for time-fractional diffusion equations in one- and two-dimensional space. A reduced CFD method is constructed with lower dimensions such that it maintains the accuracy and decreases the computational time in comparison with classical CFD method. Since the solution of time-fractional diffusion equation typically has a weak singularity near the initial time t = 0 {t=0} , the classical L1 scheme on uniform meshes may obtain a scheme with low accuracy. So, we use the L1 scheme on graded meshes for time discretization. Moreover, we provide the error estimation between the reduced CFD method based on POD and classical CFD solutions. Some numerical examples show the effectiveness and accuracy of the proposed method.

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