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A Separable Preconditioner for Time-Space Fractional Caputo-Riesz Diffusion Equations
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In this paper, we study linear systems arising from time-space fractional
Caputo-Riesz diffusion equations with time-dependent diffusion coefficients. The coefficient
matrix is a summation of a block-lower-triangular-Toeplitz matrix (temporal component)
and a block-diagonal-with-diagonal-times-Toeplitz-block matrix (spatial component).
The main aim of this paper is to propose separable preconditioners for solving
these linear systems, where a block ϵ-circulant preconditioner is used for the temporal
component, while a block diagonal approximation is used for the spatial variable. The
resulting preconditioner can be block-diagonalized in the temporal domain. Furthermore,
the fast solvers can be employed to solve smaller linear systems in the spatial
domain. Theoretically, we show that if the diffusion coefficient (temporal-dependent
or spatial-dependent only) function is smooth enough, the singular values of the preconditioned
matrix are bounded independent of discretization parameters. Numerical
examples are tested to show the performance of proposed preconditioner.
Title: A Separable Preconditioner for Time-Space Fractional Caputo-Riesz Diffusion Equations
Description:
In this paper, we study linear systems arising from time-space fractional
Caputo-Riesz diffusion equations with time-dependent diffusion coefficients.
The coefficient
matrix is a summation of a block-lower-triangular-Toeplitz matrix (temporal component)
and a block-diagonal-with-diagonal-times-Toeplitz-block matrix (spatial component).
The main aim of this paper is to propose separable preconditioners for solving
these linear systems, where a block ϵ-circulant preconditioner is used for the temporal
component, while a block diagonal approximation is used for the spatial variable.
The
resulting preconditioner can be block-diagonalized in the temporal domain.
Furthermore,
the fast solvers can be employed to solve smaller linear systems in the spatial
domain.
Theoretically, we show that if the diffusion coefficient (temporal-dependent
or spatial-dependent only) function is smooth enough, the singular values of the preconditioned
matrix are bounded independent of discretization parameters.
Numerical
examples are tested to show the performance of proposed preconditioner.
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