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Convergence analysis of high-order compact alternating direction implicit schemes for the two-dimensional time fractional diffusion equation. (English) Zbl 1264.65143

The author proposes a pair of compact alternating direction implicit schemes, I and II, for solving a two-dimensional time fractional diffusion equation describing subdiffusive phenomena with a nonhomogeneous term. Scheme I is obtained by replacing the value of the current time level with the previous one. The stability and accuracy of scheme I are investigated. Scheme I has the advantage of high accuracy with the coefficient matrix still being tridiagonal; therefore, the linear system of equations is easy to solve efficiently to a high accuracy. It is also shown that scheme I is unconditionally stable and convergent. Depending on the value of the parameter \(\gamma\in(0,1)\) in the Caputo fractional derivative \(_0^CD_t^\gamma u\) of the function \(u(x, y, t)\) making the left hand-side term of the given diffusion equation, scheme I is good for \(\gamma\in[1/2,1)\). Scheme II, which comes with a correction term, is more suitable for \(\gamma\in[0,1/2)\). The theoretical analysis is verified by some numerical experiments.

MSC:

65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35R11 Fractional partial differential equations

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