Stability and Convergence Analysis of Finite Difference Schemes for Time-Dependent Space-Fractional Diffusion Equations with Variable Diffusion Coefficients

2017 ◽  
Vol 75 (2) ◽  
pp. 1102-1127 ◽  
Author(s):  
Xue-lei Lin ◽  
Michael K. Ng ◽  
Hai-Wei Sun
Keyword(s):  
Finite Difference ◽  
Diffusion Coefficients ◽  
Fractional Diffusion ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Stability And Convergence ◽  
Finite Difference Schemes ◽  
Fractional Diffusion Equations ◽  
Variable Diffusion ◽  
Variable Diffusion Coefficients

Finite difference scheme for the time-space fractional diffusion equations

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Jianxiong Cao ◽  
Changpin Li
Keyword(s):  
Finite Difference ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Stability And Convergence ◽  
Finite Difference Schemes ◽  
Order Accuracy ◽  
Fractional Diffusion Equations ◽  
Time Space ◽  
Liouville Fractional Derivative ◽  
The Stability

AbstractIn this paper, we derive two novel finite difference schemes for two types of time-space fractional diffusion equations by adopting weighted and shifted Grünwald operator, which is used to approximate the Riemann-Liouville fractional derivative to the second order accuracy. The stability and convergence of the schemes are analyzed via mathematical induction. Moreover, the illustrative numerical examples are carried out to verify the accuracy and effectiveness of the schemes.

Multilevel Circulant Preconditioner for High-Dimensional Fractional Diffusion Equations

2016 ◽  
Vol 6 (2) ◽  
pp. 109-130 ◽  
Author(s):  
Siu-Long Lei ◽  
Xu Chen ◽  
Xinhe Zhang
Keyword(s):  
Krylov Subspace ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Low Rank ◽  
High Dimensional ◽  
Fractional Diffusion Equations ◽  
Implicit Finite Difference Scheme ◽  
Small Norm ◽  
Variable Diffusion ◽  
Variable Diffusion Coefficients

AbstractHigh-dimensional two-sided space fractional diffusion equations with variable diffusion coefficients are discussed. The problems can be solved by an implicit finite difference scheme that is proven to be uniquely solvable, unconditionally stable and first-order convergent in the infinity norm. A nonsingular multilevel circulant pre-conditoner is proposed to accelerate the convergence rate of the Krylov subspace linear system solver efficiently. The preconditoned matrix for fast convergence is a sum of the identity matrix, a matrix with small norm, and a matrix with low rank under certain conditions. Moreover, the preconditioner is practical, with an O(NlogN) operation cost and O(N) memory requirement. Illustrative numerical examples are also presented.

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