Superfast solution of real positive definite toeplitz systems
SIAM Journal on Matrix Analysis and Applications
Conjugate Gradient Methods for Toeplitz Systems
SIAM Review
Finite difference approximations for fractional advection-dispersion flow equations
Journal of Computational and Applied Mathematics
The accuracy and stability of an implicit solution method for the fractional diffusion equation
Journal of Computational Physics
Finite difference methods for two-dimensional fractional dispersion equation
Journal of Computational Physics
Journal of Computational and Applied Mathematics
Toeplitz and circulant matrices: a review
Communications and Information Theory
Numerical Approximation of a Time Dependent, Nonlinear, Space-Fractional Diffusion Equation
SIAM Journal on Numerical Analysis
Implicit finite difference approximation for time fractional diffusion equations
Computers & Mathematics with Applications
Finite difference approximations for a fractional advection diffusion problem
Journal of Computational Physics
Compact finite difference method for the fractional diffusion equation
Journal of Computational Physics
Finite difference approximations for two-sided space-fractional partial differential equations
Applied Numerical Mathematics
A direct O(Nlog2N) finite difference method for fractional diffusion equations
Journal of Computational Physics
Compact alternating direction implicit method for two-dimensional time fractional diffusion equation
Journal of Computational Physics
A radial basis functions method for fractional diffusion equations
Journal of Computational Physics
A circulant preconditioner for fractional diffusion equations
Journal of Computational Physics
Preconditioned iterative methods for fractional diffusion equation
Journal of Computational Physics
Stable multi-domain spectral penalty methods for fractional partial differential equations
Journal of Computational Physics
Operator splitting ADI schemes for pseudo-time coupled nonlinear solvation simulations
Journal of Computational Physics
Journal of Computational Physics
Hi-index | 31.48 |
Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the second-order diffusion equations. Because of the nonlocal property of fractional differential operators, the numerical methods for fractional diffusion equations often generate dense or even full coefficient matrices. Consequently, the numerical solution of these methods often require computational work of O(N^3) per time step and memory of O(N^2) for where N is the number of grid points. In this paper we develop a fast alternating-direction implicit finite difference method for space-fractional diffusion equations in two space dimensions. The method only requires computational work of O(N log^2N) per time step and memory of O(N), while retaining the same accuracy and approximation property as the regular finite difference method with Gaussian elimination. Our preliminary numerical example runs for two dimensional model problem of intermediate size seem to indicate the observations: To achieve the same accuracy, the new method has a significant reduction of the CPU time from more than 2 months and 1 week consumed by a traditional finite difference method to 1.5h, using less than one thousandth of memory the standard method does. This demonstrates the utility of the method.