Volume 14, Issue 6
An Efficient Second-Order Finite Volume ADI Method for Nonlinear Three-Dimensional Space-Fractional Reaction-Diffusion Equations

Adv. Appl. Math. Mech., 14 (2022), pp. 1400-1432.

Published online: 2022-08

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• Abstract

In this paper, a three-dimensional time-dependent nonlinear Riesz space-fractional reaction-diffusion equation is considered. First, a linearized finite volume method, named BDF-FV, is developed and analyzed via the discrete energy method, in which the space-fractional derivative is discretized by the finite volume element method and the time derivative is treated by the backward differentiation formulae (BDF). The method is rigorously proved to be convergent with second-order accuracy both in time and space with respect to the discrete and continuous $L^2$ norms. Next, by adding high-order perturbation terms in time to the BDF-FV scheme, an alternating direction implicit linear finite volume scheme, denoted as BDF-FV-ADI, is proposed. Convergence with second-order accuracy is also strictly proved under a rough temporal-spatial stepsize constraint. Besides, efficient implementation of the ADI method is briefly discussed, based on a fast conjugate gradient (FCG) solver for the resulting symmetric positive definite linear algebraic systems. Numerical experiments are presented to support the theoretical analysis and demonstrate the effectiveness and efficiency of the method for large-scale modeling and simulations.

• Keywords

Nonlinear space-fractional reaction-diffusion equation, finite volume method, ADI, convergence, efficient implementation.

65M08, 65M12, 65M15

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@Article{AAMM-14-1400, author = {Bingyin and Zhang and and 24240 and and Bingyin Zhang and Hongfei and Fu and and 24241 and and Hongfei Fu and Xueting and Liang and and 24242 and and Xueting Liang and Jun and Liu and and 24243 and and Jun Liu and Jiansong and Zhang and and 24244 and and Jiansong Zhang}, title = {An Efficient Second-Order Finite Volume ADI Method for Nonlinear Three-Dimensional Space-Fractional Reaction-Diffusion Equations}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2022}, volume = {14}, number = {6}, pages = {1400--1432}, abstract = {

In this paper, a three-dimensional time-dependent nonlinear Riesz space-fractional reaction-diffusion equation is considered. First, a linearized finite volume method, named BDF-FV, is developed and analyzed via the discrete energy method, in which the space-fractional derivative is discretized by the finite volume element method and the time derivative is treated by the backward differentiation formulae (BDF). The method is rigorously proved to be convergent with second-order accuracy both in time and space with respect to the discrete and continuous $L^2$ norms. Next, by adding high-order perturbation terms in time to the BDF-FV scheme, an alternating direction implicit linear finite volume scheme, denoted as BDF-FV-ADI, is proposed. Convergence with second-order accuracy is also strictly proved under a rough temporal-spatial stepsize constraint. Besides, efficient implementation of the ADI method is briefly discussed, based on a fast conjugate gradient (FCG) solver for the resulting symmetric positive definite linear algebraic systems. Numerical experiments are presented to support the theoretical analysis and demonstrate the effectiveness and efficiency of the method for large-scale modeling and simulations.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2021-0153}, url = {http://global-sci.org/intro/article_detail/aamm/20853.html} }
TY - JOUR T1 - An Efficient Second-Order Finite Volume ADI Method for Nonlinear Three-Dimensional Space-Fractional Reaction-Diffusion Equations AU - Zhang , Bingyin AU - Fu , Hongfei AU - Liang , Xueting AU - Liu , Jun AU - Zhang , Jiansong JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 1400 EP - 1432 PY - 2022 DA - 2022/08 SN - 14 DO - http://doi.org/10.4208/aamm.OA-2021-0153 UR - https://global-sci.org/intro/article_detail/aamm/20853.html KW - Nonlinear space-fractional reaction-diffusion equation, finite volume method, ADI, convergence, efficient implementation. AB -

In this paper, a three-dimensional time-dependent nonlinear Riesz space-fractional reaction-diffusion equation is considered. First, a linearized finite volume method, named BDF-FV, is developed and analyzed via the discrete energy method, in which the space-fractional derivative is discretized by the finite volume element method and the time derivative is treated by the backward differentiation formulae (BDF). The method is rigorously proved to be convergent with second-order accuracy both in time and space with respect to the discrete and continuous $L^2$ norms. Next, by adding high-order perturbation terms in time to the BDF-FV scheme, an alternating direction implicit linear finite volume scheme, denoted as BDF-FV-ADI, is proposed. Convergence with second-order accuracy is also strictly proved under a rough temporal-spatial stepsize constraint. Besides, efficient implementation of the ADI method is briefly discussed, based on a fast conjugate gradient (FCG) solver for the resulting symmetric positive definite linear algebraic systems. Numerical experiments are presented to support the theoretical analysis and demonstrate the effectiveness and efficiency of the method for large-scale modeling and simulations.

Bingyin Zhang, Hongfei Fu, Xueting Liang, Jun Liu & Jiansong Zhang. (2022). An Efficient Second-Order Finite Volume ADI Method for Nonlinear Three-Dimensional Space-Fractional Reaction-Diffusion Equations. Advances in Applied Mathematics and Mechanics. 14 (6). 1400-1432. doi:10.4208/aamm.OA-2021-0153
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