A general lumping analysis of a reaction system coupled with diffusion

A general lumping analysis of a reaction system coupled with diffusion is presented. This analysis can be applied to any reaction system with n species for both steady-state and transient conditions. Here we consider lumping by means of an n̂ × n constant matrix M with rank n̂ (n̂ ≤ n). When the diffusivity is independent of position and concentration, represented by the vectors r and y, respectively, it is found that under steady-state conditions, a reaction system having a species concentration vector y(r) coupled with diffusion is exactly lumpable if and only if there exist nontrivial fixed J^T[y(r)]D^-1-invariant subspaces M [here J^T[y(r)] is the transpose of the Jacobian matrix for the chemical reaction rate vector f[y(r)] and D^-1 is the inverse of the constant effective diffusivity matrix], no matter what value y(r) takes; under transient conditions, there exist simultaneously D- and J^T[y(r, t)]-invariant subspaces M. When D is a function of position or concentrations, M is simultaneously invariant to J^T(y) and D(r), D[y(r)] or D[y(r, t)]. The approach used to determine the constrained approximate lumping schemes for a nondiffusion system can also be used in a reaction-diffusion system, except that the constant basis matrices A_k's of J^T(y) are replaced by B_k = A_kD^-1 under steady-state conditions, or the extra matrix D is added under transient conditions. For n onconstant D, the basis constant matrices D_i's of D(r), D[y(r)] or D[y(r, t)] are added.