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For nonjump system, system state and control signal contain only continuous time with, while for jump system, they concern with both continuous time and discrete regime.
Hamilton [9] characterizes business cycles as periods of discrete regime shifts, i.e., recessions are characterized belonging to one regime and expansions to another in a Markov-switching process.
From these hazards, we can calculate the time to the next reaction, and the reaction type i of the discrete regime by using Gillespie's direct method [20].
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In Couette flow, the fluid displays shear thickening and thinning within a discrete shear regime.
In other words, the time until the next reaction occurs in the discrete (stochastic) regime, δτ, is not constant, and it does not match the time step of the numerical algorithm in the continuous regime, δt.
In previous work, we have extended Richardson extrapolation into the discrete stochastic regime [ 22].
The extrapolation framework can be extended to the discrete stochastic regime.
In this work, we have extended the extrapolation framework, which can increase the weak order of accuracy of existing numerical methods, to the discrete stochastic regime.
The experiments and calculations are consistent in identifying discrete crack speed regimes within which crack growth at sustained crack speeds is possible.
After these preparations, we compare the step sizes of discrete and continuous regimes, and select a smaller step size to update the simulation time as t: = t+min{δτ,δt}.
It is well-known that Langevin SDEs represent an intermediate regime between discrete stochastic chemical kinetics and the deterministic regime, arising as the number of molecules X in the system increases.
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