Highly accurate tau-leaping methods with random corrections.

We aim to construct higher order tau-leaping methods for numerically simulating stochastic chemical kinetic systems in this paper. By adding a random correction to the primitive tau-leaping scheme in each time step, we greatly improve the accuracy of the tau-leaping approximations. This gain in accuracy actually comes from the reduction in the local truncation error of the scheme in the order of tau, the marching time step size. While the local truncation error of the primitive tau-leaping method is O(tau(2)) for all moments, our Poisson random correction tau-leaping method, in which the correction term is a Poisson random variable, can reduce the local truncation error for the mean to O(tau(3)), and both Gaussian random correction tau-leaping methods, in which the correction term is a Gaussian random variable, can reduce the local truncation error for both the mean and covariance to O(tau(3)). Numerical results demonstrate that these novel methods more accurately capture crucial properties such as the mean and variance than existing methods for simulating chemical reaction systems. This work constitutes a first step to construct high order numerical methods for simulating jump processes. With further refinement and appropriately modified step-size selection procedures, the random correction methods should provide a viable way of simulating chemical reaction systems accurately and efficiently.