Correction of high-order BDF convolution adrature for fractional evolution equations

Bangti Jin; Buyang Li; Zhi Zhou

doi:10.1137/17M1118816

Correction of high-order BDF convolution adrature for fractional evolution equations

Bangti Jin, Buyang Li, Zhi Zhou

Research output: Journal article publication › Journal article › Academic research › peer-review

168 Citations (Scopus)

Abstract

We develop proper correction formulas at the starting k − 1 steps to restore the desired kth-order convergence rate of the k-step BDF convolution quadrature for discretizing evolution equations involving a fractional-order derivative in time. The desired kth-order convergence rate can be achieved even if the source term is not compatible with the initial data, which is allowed to be nonsmooth. We provide complete error estimates for the subdiffusion case α ∈ (0, 1) and sketch the proof for the diffusion-wave case α ∈ (1, 2). Extensive numerical examples are provided to illustrate the effectiveness of the proposed scheme.

Original language	English
Pages (from-to)	A3129-A3152
Journal	SIAM Journal on Scientific Computing
Volume	39
Issue number	6
DOIs	https://doi.org/10.1137/17M1118816
Publication status	Published - 1 Jan 2017

Keywords

Backward differentiation formulas
Convolution quadrature
Error estimates
Fractional evolution equation
Incompatible data
Initial correction
Nonsmooth

ASJC Scopus subject areas

Computational Mathematics
Applied Mathematics

More information

10.1137/17M1118816

http://hdl.handle.net/10397/77176

Cite this

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abstract = "We develop proper correction formulas at the starting k − 1 steps to restore the desired kth-order convergence rate of the k-step BDF convolution quadrature for discretizing evolution equations involving a fractional-order derivative in time. The desired kth-order convergence rate can be achieved even if the source term is not compatible with the initial data, which is allowed to be nonsmooth. We provide complete error estimates for the subdiffusion case α ∈ (0, 1) and sketch the proof for the diffusion-wave case α ∈ (1, 2). Extensive numerical examples are provided to illustrate the effectiveness of the proposed scheme.",

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N2 - We develop proper correction formulas at the starting k − 1 steps to restore the desired kth-order convergence rate of the k-step BDF convolution quadrature for discretizing evolution equations involving a fractional-order derivative in time. The desired kth-order convergence rate can be achieved even if the source term is not compatible with the initial data, which is allowed to be nonsmooth. We provide complete error estimates for the subdiffusion case α ∈ (0, 1) and sketch the proof for the diffusion-wave case α ∈ (1, 2). Extensive numerical examples are provided to illustrate the effectiveness of the proposed scheme.

AB - We develop proper correction formulas at the starting k − 1 steps to restore the desired kth-order convergence rate of the k-step BDF convolution quadrature for discretizing evolution equations involving a fractional-order derivative in time. The desired kth-order convergence rate can be achieved even if the source term is not compatible with the initial data, which is allowed to be nonsmooth. We provide complete error estimates for the subdiffusion case α ∈ (0, 1) and sketch the proof for the diffusion-wave case α ∈ (1, 2). Extensive numerical examples are provided to illustrate the effectiveness of the proposed scheme.

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KW - Convolution quadrature

KW - Error estimates

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Correction of high-order BDF convolution adrature for fractional evolution equations

Abstract

Keywords

ASJC Scopus subject areas

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