Tau leap methods enable efficient simulation of stochastic chemical systems. We develop new tau leap methods that consider the following issues: stiffness, fluid limit, integer-valued states, nonnegative states, and correlations among species. We propose two tau leap methods taking into account a compromise among these issues. Both tau leap methods have the implicit Euler as their fluid limit, which is accomplished by the use of a split step: the first part computes the mean update implicitly, and the second part generates a random update explicitly with the mean computed in the first part.

The first tau leap method is designed to generate physically feasible population vectors yielding integer and nonnegative population states, and we refer to it as the Minkowski-Weyl tau (MW-tau) leap method. It incorporates the Minkowski-Weyl decomposition from polyhedral theory in generating nonnegative states, and three variants are proposed to reflect the idea of the method. The second method is devised to approximate the correlations among species, and it is called the tau leap method via local linear noise approximation (LLNA-tau). It utilizes the linear noise approximation (LNA) to obtain ODEs governing the evolution of the mean and the covariance for stochastic chemical systems. LLNA-tau applies LNA to approximate the mean and the covariance for reactions within one time step. The LNA is recalculated after each time step, and the population state is updated by Gaussian random variables with the approximated mean and covariance. Analysis of consistency, stability and fluid limit of the tau leap methods are also presented.

We illustrate MW-tau and LLNA-tau by some numerical examples, and compare them with existing tau leap methods. For most cases, the MW-tau and LLNA-tau methods perform favorably.