Optimal explicit stabilized postprocessed $\tau$-leap method for the simulation of chemical kinetics

TL;DR

This paper introduces a new stabilized and postprocessed $ au$-leap method for simulating multiscale chemical kinetics, improving stability, accuracy, and efficiency in stochastic simulations.

Contribution

The paper presents PSK-$ au$-ROCK, a novel stabilized and postprocessed $ au$-leap method that enhances simulation of stiff chemical kinetic systems.

Findings

Proves stability and accuracy of PSK-$ au$-ROCK.

Demonstrates high reliability and efficiency in numerical experiments.

Outperforms existing $ au$-leap methods in simulations.

Abstract

The simulation of chemical kinetics involving multiple scales constitutes a modeling challenge (from ordinary differential equations to Markov chain) and a computational challenge (multiple scales, large dynamical systems, time step restrictions). In this paper we propose a new discrete stochastic simulation algorithm: the postprocessed second kind stabilized orthogonal $\tau$-leap Runge-Kutta method (PSK-$\tau$-ROCK). In the context of chemical kinetics this method can be seen as a stabilization of Gillespie's explicit $\tau$-leap combined with a postprocessor. The stabilized procedure allows to simulate problems with multiple scales (stiff), while the postprocessing procedure allows to approximate the invariant measure (e.g. mean and variance) of ergodic stochastic dynamical systems. We prove stability and accuracy of the PSK-$\tau$-ROCK. Numerical experiments illustrate the high reliability and efficiency of the scheme when compared to other $\tau$-leap methods.