GeCo: Geometric Conservative nonstandard schemes for biochemical systems

We generalize the nonstandard Euler and Heun schemes in order to provide explicit geometric numerical integrators for biochemical systems, here denoted as GeCo schemes, that preserve both positivity of the solutions and linear invariants. We relax the request on the order convergence of the denominator function for the first-order approximation and we let it depend on the step size also throughout the solution approximating values. The first-order variant is exact on a two-dimensional linear test problem. Moreover, we introduce a class of modified mGeCo(α) schemes and, by tuning the parameter α≥1, we improve the numerical performance of GeCo integrators on some examples taken from the literature. A numerical comparison with BBKS and mBBKS schemes, which are implicit integrators, positive and conserve linear invariants, show the gain in efficiency of both GeCo and mGeCo(α) procedures as they generate similar errors with an explicit functional form.