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arXiv · 2404.10221

On $\tau$-preconditioners for a quasi-compact difference scheme to Riesz fractional diffusion equations with variable coefficients

Zi-Hang She·Xue Zhang·Xian-Ming Gu·Stefano Serra-Capizzano
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Abstract

In the present study, we consider the preconditioned generalized minimal residual (GMRES) method for the asymmetric linear systems arising from the $d$-dimensional Riesz space fractional diffusion equations (RSFDEs). The Crank-Nicolson scheme and a quasi-compact finite difference method are used to discretize the temporal derivative and Riesz space fractional derivatives in such RSFDEs, respectively. For the $d$-dimensional discretized RSFDEs, the corresponding coefficient matrix is the sum of a product of a $d$-level block tridiagonal matrix multiplying a diagonal matrix and a $d$-level Toeplitz matrix. We develop a sine transform based preconditioner (namely $\tau$ preconditioner) to accelerate the convergence of the GMRES method. Theoretical analysis shows that the upper bound of relative residual norm of the GMRES method with the proposed preconditioner is mesh-independent, which leads to a linear convergence rate. Numerical results are presented to confirm the theoretical results regarding the preconditioned matrix and to illustrate the efficiency of the proposed preconditioner.

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Zi-Hang She, Xue Zhang, Xian-Ming Gu, Stefano Serra-Capizzano. 2024-04-16. On $\tau$-preconditioners for a quasi-compact difference scheme to Riesz fractional diffusion equations with variable coefficients. https://arxiv.org/abs/2404.10221

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