Volume 30, Issue 2
A Second-Order Scheme with Nonuniform Time Steps for a Linear Reaction-Subdiffusion Problem

Commun. Comput. Phys., 30 (2021), pp. 567-601.

Published online: 2021-05

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• Abstract

It is reasonable to assume that a discrete convolution structure dominates the local truncation error of any numerical Caputo formula because the fractional time derivative and its discrete approximation have the same convolutional form. We suggest an error convolution structure (ECS) analysis for a class of interpolation-type approximations to the Caputo fractional derivative. Our assumptions permit the use of adaptive time steps, such as is appropriate for accurately resolving the initial singularity of the solution and also certain complex behavior away from the initial time. The ECS analysis of numerical approximations has two advantages: (i) to localize (and simplify) the analysis of the approximation error of a discrete convolution formula on general nonuniform time grids; and (ii) to reveal the error distribution information in the long-time integration via the global consistency error. The core result in this paper is an ECS bound and a global consistency analysis of the nonuniform Alikhanov approximation, which is constructed at an offset point by using linear and quadratic polynomial interpolation. Using this result, we derive a sharp $L^2$-norm error estimate of a second-order Crank-Nicolson-like scheme for linear reaction-subdiffusion problems. An example is presented to show the sharpness of our analysis.

• Keywords

Caputo fractional derivative, nonuniform time mesh, error convolution structure, global consistency error, stability and convergence.

65M06, 35B65

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• TXT
@Article{CiCP-30-567, author = {Hong-Lin and Liao and and 16186 and and Hong-Lin Liao and William and McLean and and 16187 and and William McLean and Jiwei and Zhang and and 16188 and and Jiwei Zhang}, title = {A Second-Order Scheme with Nonuniform Time Steps for a Linear Reaction-Subdiffusion Problem}, journal = {Communications in Computational Physics}, year = {2021}, volume = {30}, number = {2}, pages = {567--601}, abstract = {

It is reasonable to assume that a discrete convolution structure dominates the local truncation error of any numerical Caputo formula because the fractional time derivative and its discrete approximation have the same convolutional form. We suggest an error convolution structure (ECS) analysis for a class of interpolation-type approximations to the Caputo fractional derivative. Our assumptions permit the use of adaptive time steps, such as is appropriate for accurately resolving the initial singularity of the solution and also certain complex behavior away from the initial time. The ECS analysis of numerical approximations has two advantages: (i) to localize (and simplify) the analysis of the approximation error of a discrete convolution formula on general nonuniform time grids; and (ii) to reveal the error distribution information in the long-time integration via the global consistency error. The core result in this paper is an ECS bound and a global consistency analysis of the nonuniform Alikhanov approximation, which is constructed at an offset point by using linear and quadratic polynomial interpolation. Using this result, we derive a sharp $L^2$-norm error estimate of a second-order Crank-Nicolson-like scheme for linear reaction-subdiffusion problems. An example is presented to show the sharpness of our analysis.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2020-0124}, url = {http://global-sci.org/intro/article_detail/cicp/19126.html} }
TY - JOUR T1 - A Second-Order Scheme with Nonuniform Time Steps for a Linear Reaction-Subdiffusion Problem AU - Liao , Hong-Lin AU - McLean , William AU - Zhang , Jiwei JO - Communications in Computational Physics VL - 2 SP - 567 EP - 601 PY - 2021 DA - 2021/05 SN - 30 DO - http://doi.org/10.4208/cicp.OA-2020-0124 UR - https://global-sci.org/intro/article_detail/cicp/19126.html KW - Caputo fractional derivative, nonuniform time mesh, error convolution structure, global consistency error, stability and convergence. AB -

It is reasonable to assume that a discrete convolution structure dominates the local truncation error of any numerical Caputo formula because the fractional time derivative and its discrete approximation have the same convolutional form. We suggest an error convolution structure (ECS) analysis for a class of interpolation-type approximations to the Caputo fractional derivative. Our assumptions permit the use of adaptive time steps, such as is appropriate for accurately resolving the initial singularity of the solution and also certain complex behavior away from the initial time. The ECS analysis of numerical approximations has two advantages: (i) to localize (and simplify) the analysis of the approximation error of a discrete convolution formula on general nonuniform time grids; and (ii) to reveal the error distribution information in the long-time integration via the global consistency error. The core result in this paper is an ECS bound and a global consistency analysis of the nonuniform Alikhanov approximation, which is constructed at an offset point by using linear and quadratic polynomial interpolation. Using this result, we derive a sharp $L^2$-norm error estimate of a second-order Crank-Nicolson-like scheme for linear reaction-subdiffusion problems. An example is presented to show the sharpness of our analysis.

Hong-Lin Liao, William McLean & Jiwei Zhang. (2021). A Second-Order Scheme with Nonuniform Time Steps for a Linear Reaction-Subdiffusion Problem. Communications in Computational Physics. 30 (2). 567-601. doi:10.4208/cicp.OA-2020-0124
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