Convergence Analysis of the Variational Operator Splitting Scheme for a Reaction-Diffusion System with Detailed Balance


We present a detailed convergence analysis for an operator splitting scheme proposed in [C. Liu, C. Wang, and Y. Wang, J. Comput. Phys., 436 (2021), 110253] for a reaction-diffusion system with detailed balance. The numerical scheme has been constructed based on a recently developed energetic variational formulation, in which the reaction part is reformulated in terms of the reaction trajectory, and both the reaction and diffusion parts dissipate the same free energy. The scheme is energy stable and positivity-preserving. In this paper, the detailed convergence analysis and error estimate are performed for the operator splitting scheme. The nonlinearity in the reaction trajectory equation, as well as the implicit treatment of nonlinear and singular logarithmic terms, impose challenges in numerical analysis. To overcome these difficulties, we make use of the convex nature of the logarithmic nonlinear terms. In addition, a combination of rough error estimate and refined error estimate leads to a desired bound of the numerical error in the reaction stage, in the discrete maximum norm. Furthermore, a discrete maximum principle yields the evolution bound of the numerical error function at the diffusion stage. As a direct consequence, a combination of the numerical error analysis at different stages and the consistency estimate for the operator splitting procedure results in the convergence estimate of the numerical scheme for the full reaction-diffusion system. The convergence analysis technique could be extended to a more general class of dissipative reaction mechanisms. As an example, we also consider a near-equilibrium reaction kinetics, which was derived by the linear response assumption on the reaction trajectory. Although the reaction rate is more complicated in terms of concentration variables, we show that the numerical approach and the convergence analysis also work in this case.


  1. reaction-diffusion system
  2. energetic variational formulation
  3. operator splitting scheme
  4. positivity preserving
  5. optimal rate convergence analysis
  6. rough error estimate and refined error estimate

MSC codes

  1. 35K35
  2. 35K55
  3. 49J40
  4. 65M06
  5. 65M12

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Information & Authors


Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 781 - 803
ISSN (online): 1095-7170


Submitted: 20 May 2021
Accepted: 27 January 2022
Published online: 13 April 2022


  1. reaction-diffusion system
  2. energetic variational formulation
  3. operator splitting scheme
  4. positivity preserving
  5. optimal rate convergence analysis
  6. rough error estimate and refined error estimate

MSC codes

  1. 35K35
  2. 35K55
  3. 49J40
  4. 65M06
  5. 65M12



Funding Information

Illinois Institute of Technology
National Science Foundation : DMS-1759536, DMS-1950868, DMS-2012669, DMS-1719854, DMS-2012634

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