The stochastic simulation algorithm (SSA), proposed by Gillespie, is a cardinal simulation method for the chemical kinetics. Because the SSA simulates every reaction event, the amount of the computational time is huge when models have many reaction channels and species. This drawback motivates many attempts to improve the efficiency with the accuracy. The existing "implicit tau-leaping" procedure attempts to accelerate the exact SSA especially for stiff systems. The implicit tau-leaping method uses an implicit discretization for the mean, together with an explicit discretization of the Poisson "noise". It is therefore a partially implicit method.
In this paper we propose three fully implicit tau-leaping methods that treat implicitly both the mean part and the variance of the Poisson variables. The three methods considered below are the backward Euler for the mean and backward Euler for the variance of the Poisson variables, trapezoidal for both the mean and the variance of the Poisson variables, and backward Euler for the mean and trapezoidal for the variance of the Poisson variables. These methods are motivated by the theory of weakly convergent discretizations of stochastic differential equations. Numerical results demonstrate the performance of the new fully implicit methods.
[1]
A. Arkin,et al.
Stochastic mechanisms in gene expression.
,
1997,
Proceedings of the National Academy of Sciences of the United States of America.
[2]
Muruhan Rathinam,et al.
Consistency and Stability of Tau-Leaping Schemes for Chemical Reaction Systems
,
2005,
Multiscale Model. Simul..
[3]
D. Gillespie.
Approximate accelerated stochastic simulation of chemically reacting systems
,
2001
.
[4]
Linda R. Petzold,et al.
Improved leap-size selection for accelerated stochastic simulation
,
2003
.
[5]
D. Gillespie.
A rigorous derivation of the chemical master equation
,
1992
.
[6]
Hong Li,et al.
Efficient formulation of the stochastic simulation algorithm for chemically reacting systems.
,
2004,
The Journal of chemical physics.
[7]
Muruhan Rathinam,et al.
Stiffness in stochastic chemically reacting systems: The implicit tau-leaping method
,
2003
.
[8]
E. Renshaw,et al.
STOCHASTIC DIFFERENTIAL EQUATIONS
,
1974
.
[9]
D. Gillespie.
Exact Stochastic Simulation of Coupled Chemical Reactions
,
1977
.