  In this paper, we propose and analyze a positivity-preserving, energy stable
numerical scheme for certain type 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 the diffusion parts dissipate the same free
energy opens a path of 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.
To our best knowledge, it is the first time to report an
energy-dissipation-law-based operator splitting scheme to a nonlinear PDE with
variational structures. Several numerical examples are presented to demonstrate
the robustness of the proposed operator splitting scheme.
