  We present a detailed convergence analysis for an operator splitting scheme
proposed in [C. Liu et al.,J. Comput. Phys., 436, 110253, 2021] 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, which are treated
implicitly in the chemical reaction stage. In addition, a combination of a
rough error estimate and a 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 results in the convergence
estimate of the numerical scheme for the full reaction-diffusion system.
