scholarly journals A class of second order difference approximations for solving space fractional diffusion equations

2015 ◽  
Vol 84 (294) ◽  
pp. 1703-1727 ◽  
Author(s):  
WenYi Tian ◽  
Han Zhou ◽  
Weihua Deng
Keyword(s):  
Fractional Diffusion ◽  
Second Order ◽  
Diffusion Equations ◽  
Order Difference ◽  
Fractional Diffusion Equations ◽  
Difference Approximations

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.

Second Order Accuracy Finite Difference Methods for Fractional Diffusion Equations

2013 ◽  
Author(s):  
Yuki Takeuchi ◽  
Reiji Suda
Keyword(s):  
Finite Difference ◽  
Diffusion Equation ◽  
Finite Difference Methods ◽  
Fractional Diffusion ◽  
Second Order ◽  
Fractional Diffusion Equation ◽  
Diffusion Equations ◽  
Order Accuracy ◽  
Fractional Diffusion Equations ◽  
Difference Methods

Finite difference methods for fractional differential equation are ever proposed. However, precise error orders have not been analyzed for the methods higher than first order accuracy. This paper proposes a few finite difference methods for fractional diffusion equations and shows our methods have second order accuracy under the conditions that the solution functions have higher order than second order at boundaries. In addition, we show that the accuracy may decrease in the case that the solution functions have lower order than second order at boundaries when we use second order accuracy scheme. In this paper, we treat schemes based on Grunwald-Letnikov definition and apply them to three kinds of fractional diffusion equations using Riemann-Liouville derivative operator including time-fractional diffusion equation, space-fractional diffusion equation and time-space-fractional diffusion equation. Finally, we show the simulation results which indicate that our methods are stable and have successfully second order accuracy under the assumed conditions.

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