TY - JOUR
T1 - A structure-preserving, operator splitting scheme for reaction-diffusion equations with detailed balance
AU - Liu, Chun
AU - Wang, Cheng
AU - Wang, Yiwei
N1 - Funding Information:
This work is partially supported by the National Science Foundation (USA) grants NSF DMS-1759536 , NSF DMS-1950868 (C. Liu, Y. Wang) and NSF DMS-2012669 (C. Wang). Y. Wang would also like to thank Department of Applied Mathematics at Illinois Institute of Technology for their generous support and for a stimulating environment.
Publisher Copyright:
© 2021 Elsevier Inc.
PY - 2021/7/1
Y1 - 2021/7/1
N2 - In this paper, we propose and analyze a positivity-preserving, energy stable numerical scheme for a certain type of reaction-diffusion systems involving the Law of Mass Action with the detailed balance condition. The numerical scheme is constructed based on a recently developed energetic variational formulation, in which the reaction part is reformulated in terms of reaction trajectories. The fact that both the reaction and diffusion parts dissipate the same free energy opens a path of designing an energy stable, operator splitting scheme for these systems. At the reaction stage, we solve equations of reaction trajectories by treating all the logarithmic terms in the reformulated form implicitly due to their convex nature. The positivity-preserving property and unique solvability can be theoretically proved, based on the singular behavior of the logarithmic function around the limiting value. Moreover, the energy stability of this scheme at the reaction stage can be proved by a careful convexity analysis. Similar techniques are used to establish the positivity-preserving property and energy stability for the standard semi-implicit solver at the diffusion stage. As a result, a combination of these two stages leads to a positivity-preserving and energy stable numerical scheme for the original reaction-diffusion system. Several numerical examples are presented to demonstrate the robustness of the proposed operator splitting scheme.
AB - In this paper, we propose and analyze a positivity-preserving, energy stable numerical scheme for a certain type of reaction-diffusion systems involving the Law of Mass Action with the detailed balance condition. The numerical scheme is constructed based on a recently developed energetic variational formulation, in which the reaction part is reformulated in terms of reaction trajectories. The fact that both the reaction and diffusion parts dissipate the same free energy opens a path of designing an energy stable, operator splitting scheme for these systems. At the reaction stage, we solve equations of reaction trajectories by treating all the logarithmic terms in the reformulated form implicitly due to their convex nature. The positivity-preserving property and unique solvability can be theoretically proved, based on the singular behavior of the logarithmic function around the limiting value. Moreover, the energy stability of this scheme at the reaction stage can be proved by a careful convexity analysis. Similar techniques are used to establish the positivity-preserving property and energy stability for the standard semi-implicit solver at the diffusion stage. As a result, a combination of these two stages leads to a positivity-preserving and energy stable numerical scheme for the original reaction-diffusion system. Several numerical examples are presented to demonstrate the robustness of the proposed operator splitting scheme.
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U2 - 10.1016/j.jcp.2021.110253
DO - 10.1016/j.jcp.2021.110253
M3 - Article
AN - SCOPUS:85102889873
SN - 0021-9991
VL - 436
JO - Journal of Computational Physics
JF - Journal of Computational Physics
M1 - 110253
ER -