Communications in Mathematical Sciences

Volume 9 (2011)

Number 4

The weak convergence analysis of tau-leaping methods: revisited

Pages: 965 – 996



Yucheng Hu (School of Mathematical Sciences, Peking University, Beijing, China)

Tiejun Li (School of Mathematical Sciences, Peking University, Beijing, China)

Bin Min (School of Mathematical Sciences, Peking University, Beijing, China)


There are two scalings for the convergence analysis of tau-leaping methods in the literature. This paper attempts to resolve this debate in the paper. We point out the shortcomings of both scalings. We systematically develop the weak Ito-Taylor expansion based on the infinitesimal generator of the chemical kinetic system and generalize the rooted tree theory for ODEs and SDEs driven by Brownian motion to rooted directed graph theory for the jump processes. We formulate the local truncation error analysis based on the large volume scaling. We find that even in this framework the midpoint tau-leaping does not improve the weak local order for the covariance compared with the explicit tau-leaping. We propose a procedure to explain the numerical order behavior by abandoning the dependence on the volume constant V from the leading error term. The numerical examples validate our arguments. We also give a general global weak convergence analysis for the explicit tau-leaping type methods in the large volume scaling.


chemical reaction kinetics, large volume scaling, convergence analysis, rooted tree theory

2010 Mathematics Subject Classification

65C30, 65G50, 92C45

Published 29 July 2011