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She, Zi-Hang; Qiu, Li-Min; Qu, Wei

An unconditionally convergent RSCSCS iteration method for Riesz space fractional diffusion equations with variable coefficients. (English) Zbl 1540.65327

Math. Comput. Simul. 203, 633-646 (2023).
Summary: In this paper, a respectively scaled circulant and skew-circulant splitting (RSCSCS) iteration method is employed to solve the Toeplitz-like linear systems arising from time-dependent Riesz space fractional diffusion equations with variable coefficients. The RSCSCS iteration method is shown to be convergent unconditionally by a novel technique, and only requires computational costs of \(\mathcal{O} (N \log N)\) with \(N\) denoting the number of interior mesh points in space. In theory, we obtain an upper bound for the convergence factor of the RSCSCS iteration method and discuss the optimal value of its iteration parameter that minimizes the corresponding upper bound. Meanwhile, we also design a fast induced RSCSCS preconditioner to accelerate the convergence rate of the Krylov subspace iteration method likes generalized minimal residual (GMRES) method. Numerical results are presented to show that the efficiencies of our proposed RSCSCS iteration method and the preconditioned GMRES method with the RSCSCS preconditioner.

Cited in 2 Documents

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35R11 Fractional partial differential equations

Keywords:

Riesz space fractional diffusion equations; circulant and skew-circulant splitting; unconditional convergence; preconditioning
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References:

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