On a positive-preserving, energy-stable numerical scheme to mass-action kinetics with detailed balance
Chun Liu, Cheng Wang, Yiwei Wang
TL;DR
This work develops a structure-preserving numerical method for mass-action kinetics with detailed balance by recasting the dynamics as a gradient-flow in reaction-trajectory space. The first-order semi-implicit discretization reduces to a convex variational problem $J^n(R) = d_R^2(R, R^n) + F[c(R)]$ whose minimizer yields the next step while enforcing positivity and energy dissipation, ensuring unconditional energy stability. The authors prove positivity-preserving and unique solvability for the multi-reaction case ($M\le N$, ${\rm rank}(S) = M$), establishing convexity of $J^n$ and showing the minimizer cannot lie on the boundary; they illustrate the arguments in a $M=2$, $N=4$ setting. This extends previous results from $M=1$ to general multi-reaction networks under detailed balance and provides a rigorous foundation for long-time, physically faithful simulations of CRNs using trajectory-based structure-preserving discretizations.
Abstract
In this paper, we provide a detailed theoretical analysis of the numerical scheme introduced in J. Comput. Phys. 436 (2021) 110253 for the reaction kinetics of a class of chemical reaction networks that satisfies detailed balance condition. In contrast to conventional numerical approximations, which are typically constructed based on ordinary differential equations (ODEs) for the concentrations of all involved species, the scheme is developed using the equations of reaction trajectories, which can be viewed as a generalized gradient flow of physically relevant free energy. The unique solvability, positivity-preserving, and energy-stable properties are proved for the general case involving multiple reactions, under a mild condition on the stoichiometric matrix.