Compact alternating direction implicit method for two-dimensional time fractional diffusion equation. (English) Zbl 1242.65158
Summary: High-order compact finite difference scheme with operator splitting technique for solving two-dimensional time fractional diffusion equation is considered in this paper. A Grünwald-Letnikov approximation is used for the Riemann-Liouville time derivative, and the second order spatial derivatives are approximated by the compact finite differences to obtain a fully discrete implicit scheme. Alternating direction implicit (ADI) method is used to split the original problem into two separate one-dimensional problems. The local truncation error is analyzed and the stability is discussed by the Fourier method. The proposed scheme is suitable when the order of the time fractional derivative \(\gamma \) lies in the interval \(\gamma \in (0,\frac 12)\). A correction term is added to maintain high accuracy when \([\frac 12,1)\). Numerical results are provided to verify the accuracy and efficiency of the proposed algorithm.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 35K05 | Heat equation |
| 35R11 | Fractional partial differential equations |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
Keywords:
fractional diffusion equation; compact scheme; finite difference; alternating direction iterative method; stability; Fourier analysis; truncation error; numerical results; algorithmReferences:
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