Volume 12, Issue 3
High-Order Local Absorbing Boundary Conditions for Fractional Evolution Equations on Unbounded Strips

Adv. Appl. Math. Mech., 12 (2020), pp. 664-693.

Published online: 2020-04

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• Abstract

The study of this paper is two-fold. On the one hand, we reduce the subdiffusion ($0<\alpha<1$) and diffusion-wave ($1<\alpha<2$) problems on unbounded strips to initial boundary value problems (IBVPs) by deriving high-order local artificial boundary conditions (ABCs). After that, the IBVPs with our high-order local ABCs are proved to be stable in the L2-norm. On the other hand, unconditionally stable schemes are constructed to numerically solve the IBVPs by using L1 approximation to discretize the temporal derivative and using finite difference methods to discretize the spatial derivative. We provide the complete error estimates for the subdiffusion case and sketch the proof for the diffusion-wave case. To further reduce the computational and storage cost for the evaluation of the fractional derivatives, the fast algorithm presented in [14] is employed for the case of $0<\alpha<1$ and a similar algorithm for the case of  $1<\alpha<2$ is first introduced in this article. Numerical examples are provided to verify the effectiveness and performance of our ABCs and numerical methods.

• Keywords

Subdiffusion equations, diffusion-wave equations, anomalous diffusion, artificial boundary methods, fast algorithms, high-order local absorbing boundary conditions.

35R11, 34A08, 65D20, 65N06, 65N12, 65N15

hxdong@csrc.ac.cn (Haixia Dong)

wangm@csrc.ac.cn (Miao Wang)

yindongsheng@tsinghua.edu.cn (Dongsheng Yin)

go9563@wayne.edu (Qian Zhang)

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• RIS
• TXT
@Article{AAMM-12-664, author = {Dong , Haixia and Wang , Miao and Yin , Dongsheng and Zhang , Qian}, title = {High-Order Local Absorbing Boundary Conditions for Fractional Evolution Equations on Unbounded Strips}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2020}, volume = {12}, number = {3}, pages = {664--693}, abstract = {

The study of this paper is two-fold. On the one hand, we reduce the subdiffusion ($0<\alpha<1$) and diffusion-wave ($1<\alpha<2$) problems on unbounded strips to initial boundary value problems (IBVPs) by deriving high-order local artificial boundary conditions (ABCs). After that, the IBVPs with our high-order local ABCs are proved to be stable in the L2-norm. On the other hand, unconditionally stable schemes are constructed to numerically solve the IBVPs by using L1 approximation to discretize the temporal derivative and using finite difference methods to discretize the spatial derivative. We provide the complete error estimates for the subdiffusion case and sketch the proof for the diffusion-wave case. To further reduce the computational and storage cost for the evaluation of the fractional derivatives, the fast algorithm presented in [14] is employed for the case of $0<\alpha<1$ and a similar algorithm for the case of  $1<\alpha<2$ is first introduced in this article. Numerical examples are provided to verify the effectiveness and performance of our ABCs and numerical methods.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2019-0115}, url = {http://global-sci.org/intro/article_detail/aamm/16419.html} }
TY - JOUR T1 - High-Order Local Absorbing Boundary Conditions for Fractional Evolution Equations on Unbounded Strips AU - Dong , Haixia AU - Wang , Miao AU - Yin , Dongsheng AU - Zhang , Qian JO - Advances in Applied Mathematics and Mechanics VL - 3 SP - 664 EP - 693 PY - 2020 DA - 2020/04 SN - 12 DO - http://doi.org/10.4208/aamm.OA-2019-0115 UR - https://global-sci.org/intro/article_detail/aamm/16419.html KW - Subdiffusion equations, diffusion-wave equations, anomalous diffusion, artificial boundary methods, fast algorithms, high-order local absorbing boundary conditions. AB -

The study of this paper is two-fold. On the one hand, we reduce the subdiffusion ($0<\alpha<1$) and diffusion-wave ($1<\alpha<2$) problems on unbounded strips to initial boundary value problems (IBVPs) by deriving high-order local artificial boundary conditions (ABCs). After that, the IBVPs with our high-order local ABCs are proved to be stable in the L2-norm. On the other hand, unconditionally stable schemes are constructed to numerically solve the IBVPs by using L1 approximation to discretize the temporal derivative and using finite difference methods to discretize the spatial derivative. We provide the complete error estimates for the subdiffusion case and sketch the proof for the diffusion-wave case. To further reduce the computational and storage cost for the evaluation of the fractional derivatives, the fast algorithm presented in [14] is employed for the case of $0<\alpha<1$ and a similar algorithm for the case of  $1<\alpha<2$ is first introduced in this article. Numerical examples are provided to verify the effectiveness and performance of our ABCs and numerical methods.

Haixia Dong, Miao Wang, Dongsheng Yin & Qian Zhang. (2020). High-Order Local Absorbing Boundary Conditions for Fractional Evolution Equations on Unbounded Strips. Advances in Applied Mathematics and Mechanics. 12 (3). 664-693. doi:10.4208/aamm.OA-2019-0115
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