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<p>In this paper, we study numerical methods for the solution of all-at-once linear systems arising in multi-term time-fractional high-dimensional diffusion equations (MT-TFDEs). By imposing a suitable ordering on the unknowns and capitalizing on the property that a symmetric tridiagonal Toeplitz matrix is diagonalizable via a discrete sine transform matrix, we achieve complete decoupling of the all-at-once linear system into a set of independent linear subsystems, each characterized by a lower triangular Toeplitz coefficient matrix. Then, by approximating each lower triangular Toeplitz block with an <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({r}\)</EquationSource> </InlineEquation>-circulant matrix, we construct <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({r}\)</EquationSource> </InlineEquation>-circulant preconditioners and prove that the spectra of the preconditioned matrices cluster around unity. Numerical examples are presented to validate the performance of the proposed preconditioners.&#xa0;&#xa0;&#xa0;&#xa0;&#xa0;&#xa0;&#xa0;&#xa0;</p>

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Efficient solvers for all-at-once systems in multi-term time-fractional diffusion equations

  • 期刊论文

  • 发表日期：

    2026年5月19日

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  • Nan Lin,
  • Fu-Rong Lin

摘要

In this paper, we study numerical methods for the solution of all-at-once linear systems arising in multi-term time-fractional high-dimensional diffusion equations (MT-TFDEs). By imposing a suitable ordering on the unknowns and capitalizing on the property that a symmetric tridiagonal Toeplitz matrix is diagonalizable via a discrete sine transform matrix, we achieve complete decoupling of the all-at-once linear system into a set of independent linear subsystems, each characterized by a lower triangular Toeplitz coefficient matrix. Then, by approximating each lower triangular Toeplitz block with an \({r}\) -circulant matrix, we construct \({r}\) -circulant preconditioners and prove that the spectra of the preconditioned matrices cluster around unity. Numerical examples are presented to validate the performance of the proposed preconditioners.