(622b) An Adaptive Time Stepping Scheme for Systems of Stochastic Differential Equations with Multiple Multiplicative Noise. Chemical Langevin Equations, a Proof of Concept
Stochastic differential equations (SDEs) can be numerically integrated in a similar fashion as ordinary differential equations (ODEs). There are few similarities between numerical schemes for SDEs and ODEs mainly because the theory behind SDEs becomes complicated and differs from that for ODEs for adaptive, higher order and implicit integration methods. Similar to ODEs, multiple time scales in the underlying models cause a system of SDEs to become mathematically stiff. Conventional fixed step methods, in both the stochastic and the deterministic regime, require a small time step for integrating stiff systems. Therefore become computationally slow. In addition, stiffness may arise during some parts of the simulation allowing for a larger time step in the remaining time interval.
In this work we create an adaptive scheme that deals with stiffness in SDEs by appropriately adjusting the time step, decreases the time step of the numerical integration when stiffness exists, but increases it when the system is no longer stiff. This will increase the computational efficiency of the integrator but also add stability. The variable step size algorithm is based upon existing components found in the literature. We use the Milstein Method and combine it with a binary adaptive time stepping scheme. Local error criteria are used in the decisive procedure of increasing or decreasing the time step.
The effectiveness of the proposed scheme is examined through a series of computational experiments based on systems of Chemical Langevin equations (CLEs), which are Itô SDEs with multiple multiplicative noise.