Stochastic linear multistep methods for the simulation of chemical kinetics

Area of Science:

• Computational mathematics
• Stochastic differential equations
• Numerical analysis

Background:

• The τ-leaping framework is a powerful tool for simulating jump-diffusion processes.
• Existing methods often struggle to incorporate past information efficiently.
• Accurate and efficient numerical methods are crucial for understanding complex stochastic systems.

Purpose of the Study:

• To introduce novel Stochastic Adams-Bashforth (SAB) and Stochastic Adams-Moulton (SAM) methods.
• To extend the τ-leaping framework by incorporating past information.
• To analyze the convergence properties of these new methods.

Main Methods:

• Development of k-step SAB and predictor-corrector SAM methods.
• Utilizing a Θ-trapezoidal τ-leap method of weak order two as a starting procedure.
• Analytical derivation of convergence orders for linear problems.
• Numerical testing on both linear and non-linear systems.

Main Results:

• The k-step SAB method (k ≥ 3) achieves weak order three in both mean and correlation.
• The predictor-corrector SAM method demonstrates weak order three in the mean.
• The SAM method exhibits weak order one in the correlation.
• Convergence results were validated through numerical simulations.

Conclusions:

• The proposed SAB and SAM methods offer effective extensions to the τ-leaping framework.
• SAB methods provide higher-order accuracy in both mean and correlation.
• The study highlights the importance of method choice for desired accuracy in stochastic simulations.