Hu Yucheng, Li Tiejun
Laboratory of Mathematics and Applied Mathematics and School of Mathematical Sciences, Peking University, Beijing 100871, People's Republic of China.
J Chem Phys. 2009 Mar 28;130(12):124109. doi: 10.1063/1.3091269.
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.
本文旨在构建高阶τ-跳跃方法，用于对随机化学动力学系统进行数值模拟。通过在每个时间步对原始的τ-跳跃格式添加一个随机修正项，我们极大地提高了τ-跳跃近似的精度。这种精度的提升实际上源于该格式在τ（即步长）量级上局部截断误差的减小。虽然原始τ-跳跃方法对于所有矩的局部截断误差为O(τ(2))，但我们的泊松随机修正τ-跳跃方法（其修正项为泊松随机变量）能将均值的局部截断误差降低到O(τ(3))，而两种高斯随机修正τ-跳跃方法（其修正项为高斯随机变量）能将均值和协方差的局部截断误差都降低到O(τ(3))。数值结果表明，与现有模拟化学反应系统的方法相比，这些新方法能更准确地捕捉诸如均值和方差等关键性质。这项工作是构建用于模拟跳跃过程的高阶数值方法的第一步。通过进一步优化和适当修改步长选择程序，随机修正方法应该能提供一种准确且高效地模拟化学反应系统的可行方法。
