Positive and conservative schemes for mass action kinetics

Some implicit schemes for the discretization of mass action kinetics are presented and discussed. Although all the schemes are classical, the solution procedure takes great advantage of the peculiar structure of the ODE. All the schemes maintain linear first integrals such as atomic mass conservation. The main features of the first-order scheme is positivity preservation for arbitrarily large time steps, while for the higher order schemes, there is numerical evidence that this property is maintained with fairly large time steps. For the first-order scheme, the existence of a nonnegative solution of the algebraic system arising in the advancing step is proved for arbitrarily large time steps, while uniqueness is proved for limited time steps. Finally, an efficient procedure is presented for solving the nonlinear systems involved in the time step. These systems are solved without the use of Jacobian matrices or their approximations, but by the repeated inversion of M-matrices, a procedure which is both easy and fast, considerably simplifying and accelerating the computer implementation of the schemes.