Splitting preconditioning based on sine transform for time-dependent Riesz space fractional diffusion equations
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Splitting preconditioning based on sine transform for time-dependent Riesz space fractional diffusion equations Xin Lu1 · Zhi-Wei Fang1 · Hai-Wei Sun2 Received: 29 August 2020 / Revised: 22 October 2020 / Accepted: 24 October 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract We study the sine-transform-based splitting preconditioning technique for the linear systems arising in the numerical discretization of time-dependent one dimensional and two dimensional Riesz space fractional diffusion equations. Those linear systems are Toeplitz-like. By making use of diagonal-plus-Toeplitz splitting iteration technique, a sine-transform-based splitting preconditioner is proposed to accelerate the convergence rate efficiently when the Krylov subspace method is implemented. Theoretically, we prove that the spectrum of the preconditioned matrix of the proposed method is clustering around 1. In practical computations, by the fast sine transform the computational complexity at each time level can be done in O(n log n) operations where n is the matrix size. Numerical examples are presented to illustrate the effectiveness of the proposed algorithm. Keywords Riesz space fractional diffusion equations · Shifted Grünwald discretization · Symmetric positive definite Toeplitz matrix · Sine-transform-based splitting preconditioner · GMRES method Mathematics Subject Classification 65F08 · 65F10 · 65M06 · 65M22
The research is funded in part by the National Natural Science Foundation of China under Grant 12001104, the Guangdong Basic and Applied Basic Research Foundation under Grant Nos. 2019A1515110279; 2019A1515110893; 2019A1515010789, and the Science and Technology Development Fund, Macau SAR (file no. 0118/2018/A3), MYRG2018-00015-FST from University of Macau.
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Zhi-Wei Fang [email protected] Xin Lu [email protected] Hai-Wei Sun [email protected]
1
School of Mathematics and Big Data, Foshan University, Foshan 528000, Guangdong, China
2
Department of Mathematics, University of Macau, Macao, China
123
X. Lu et al.
1 Introduction We first introduce the 1D initial-boundary value problem of the time-dependent Riesz spatial fractional diffusion equations as following [3]: ⎧ ∂u(x, t) ∂ γ u(x, t) ⎪ ⎪ d(x, t) − = f (x, t), (x, t) ∈ (x L , x R ) × [0, +∞), ⎪ ⎨ ∂t ∂|x|γ (1.1) u(x L , t) = u(x R , t) = 0, t ∈ [0, +∞), ⎪ ⎪ ⎪ ⎩ u(x, 0) = u 0 (x), x ∈ [x L , x R ], where f (x, t) is the source term and d(x, t) > 0 is the coefficient function. Here ∂ γ u(x,t) denotes the Riesz fractional derivative [23] of order γ ∈ (1, 2) and is defined γ ∂|x| by ∂ γ u(x, t) γ γ = σγ ( x L Dx + x Dx R )u(x, t), ∂|x|γ where σγ = − 2 cos(1 π γ ) > 0, and 2
γ x L Dx
γ
u(x, t), x Dx R u(x, t) are left- and right-sided
Riemann–Liouville fractional derivatives, respectively, defined by γ x L Dx
∂2 1 u(x, t) = (2 − γ ) ∂ x 2
∂2 1 γ x D x R u(x, t) = (2 − γ ) ∂ x 2
x
xL xR x
u(ξ, t) dξ, (x − ξ )γ −1 u(ξ, t) dξ. (ξ − x)γ −1
In above definitions, (·) denotes the gamma function. Fractional diffusion equations
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