  We present a novel computational framework for diffusive-reactive systems
that satisfies the non-negative constraint and maximum principles on general
computational grids. The governing equations for the concentration of reactants
and product are written in terms of tensorial diffusion-reaction equations. %
We restrict our studies to fast irreversible bimolecular reactions. If one
assumes that the reaction is diffusion-limited and all chemical species have
the same diffusion coefficient, one can employ a linear transformation to
rewrite the governing equations in terms of invariants, which are unaffected by
the reaction. This results in two uncoupled tensorial diffusion equations in
terms of these invariants, which are solved using a novel non-negative solver
for tensorial diffusion-type equations. The concentrations of the reactants and
the product are then calculated from invariants using algebraic manipulations.
The novel aspect of the proposed computational framework is that it will always
produce physically meaningful non-negative values for the concentrations of all
chemical species. Several representative numerical examples are presented to
illustrate the robustness, convergence, and the numerical performance of the
proposed computational framework. We will also compare the proposed framework
with other popular formulations. In particular, we will show that the Galerkin
formulation (which is the standard single-field formulation) does not produce
reliable solutions, and the reason can be attributed to the fact that the
single-field formulation does not guarantee non-negative solutions. We will
also show that the clipping procedure (which produces non-negative solutions
but is considered as a variational crime) does not give accurate results when
compared with the proposed computational framework.
