Applied Mathematics and Computation
SIAM Journal on Numerical Analysis
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
A second-order accurate numerical method for the two-dimensional fractional diffusion equation
Journal of Computational Physics
Numerical Approximation of a Time Dependent, Nonlinear, Space-Fractional Diffusion Equation
SIAM Journal on Numerical Analysis
A Fourier method for the fractional diffusion equation describing sub-diffusion
Journal of Computational Physics
Matrix approach to discrete fractional calculus II: Partial fractional differential equations
Journal of Computational Physics
Compact finite difference method for the fractional diffusion equation
Journal of Computational Physics
SIAM Journal on Numerical Analysis
A Space-Time Spectral Method for the Time Fractional Diffusion Equation
SIAM Journal on Numerical Analysis
Numerical simulations of 2D fractional subdiffusion problems
Journal of Computational Physics
A direct O(Nlog2N) finite difference method for fractional diffusion equations
Journal of Computational Physics
A compact finite difference scheme for the fractional sub-diffusion equations
Journal of Computational Physics
Numerical approaches to fractional calculus and fractional ordinary differential equation
Journal of Computational Physics
Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation
Journal of Computational Physics
Compact alternating direction implicit method for two-dimensional time fractional diffusion equation
Journal of Computational Physics
Journal of Computational Physics
A high-order compact exponential scheme for the fractional convection-diffusion equation
Journal of Computational and Applied Mathematics
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High-order compact finite difference method with operator-splitting technique for solving the two dimensional time fractional diffusion equation is considered in this paper. The Caputo derivative is evaluated by the L1 approximation, and the second order derivatives with respect to the space variables are approximated by the compact finite differences to obtain fully discrete implicit schemes. Alternating Direction Implicit (ADI) method is used to split the original problem into two separate one dimensional problems. One scheme is given by replacing the unknowns by the values on the previous level directly and a correction term is added for another scheme. Theoretical analysis for the first scheme is discussed. The local truncation error is analyzed and the stability is proved by the Fourier method. Using the energy method, the convergence of the compact finite difference scheme is proved. Numerical results are provided to verify the accuracy and efficiency of the two proposed algorithms. For the order of the temporal derivative lies in different intervals $\left(0,\frac{1}{2}\right)$ or $\left[\frac{1}{2},1\right)$ , corresponding appropriate scheme is suggested.