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Fast solvers for finite difference scheme of two-dimensional time-space fractional differential equations. (English) Zbl 1442.65162
Summary: Generally, solving linear systems from finite difference alternating direction implicit scheme of two-dimensional time-space fractional differential equations with Gaussian elimination requires \(\mathcal{O}\left({NM}_1M_2\left({M_1^2}+{M_2^2}+NM_1M_2\right)\right)\) complexity and \(\mathcal{O}\left({N{M_1^2}{M_2^2}}\right)\) storage, where \(N\) is the number of temporal unknown and \(M_1, M_2\) are the numbers of spatial unknown in \(x, y\) directions respectively. By exploring the structure of the coefficient matrix in fully coupled form, it possesses block lower-triangular Toeplitz structure and its blocks are block-dense Toeplitz matrices with dense-Toeplitz blocks. Based on this special structure and cooperating with time-marching or divide-and-conquer technique, two fast solvers with storage \(\mathcal{O}\left ({NM}_1M_2\right)\) are developed. The complexity for the fast solver via time-marching is \(\mathcal{O}\left({NM}_1M_2\left(N+\log\left(M_1M_2\right)\right)\right)\) and the one via divide-and-conquer technique is \(\mathcal{O}\left({NM}_1M_2\left(\log^2 N+\log\left(M_1M_2\right)\right)\right)\). It is worth to remark that the proposed solvers are not lossy. Some discussions on achieving convergence rate for smooth and non-smooth solutions are given. Numerical results show the high efficiency of the proposed fast solvers.
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M22 Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs
35R11 Fractional partial differential equations
65Y20 Complexity and performance of numerical algorithms
65F05 Direct numerical methods for linear systems and matrix inversion
Full Text: DOI
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