A chemical reaction network is a chemical system involving multiple reactions and chemical species. The simplest stochastic models of such networks treat the system as a continuous time Markov chain with the state being the number of molecules of each species and with reactions modeled as possible transitions of the chain. In this paper we provide a general framework for understanding the weak error of numerical approximation techniques in this setting. For such models, there is typically a wide variation in scales in that the different species and reaction rates vary over several orders of magnitude. Quantifying how different numerical approximation techniques behave in this settingtherefore requires that these scalings be taken into account in an appropriate manner . We quantify how the error of  different methods depends upon both the natural scalings within a given system, and with the step-size of the numerical method. We show that Euler's method, also called explicit tau-leaping, acts as an order one method, in that the error decreases linearly with the step-size, and that the approximate midpoint method acts as either an order one or two method, depending on the relation between the time-step and the scalings in the system. Further, we introduce a new algorithm in this setting, the weak trapezoidal algorithm, which has been studied previously only in the diffusive setting , and prove that it is second order accurate in the size of the time discretization, making it the first of its kind. In the multi-scale setting it is typically an extremely difficult task to perform approximations, such as Langevin approximations or law of large number type arguments, to simplify or reduce a system. Therefore, numerical methods oftentimes are the only reasonable means by which such models can be understood in real time .
A chemical reaction network is a chemical system involving multiple reactions and chemical species. The simplest stochastic models of such networks treat the system as a continuous time Markov chain with the state being the number of molecules of each species and with reactions modeled as possible transitions of the chain. For such models, there is typically a wide variation in temporal and other quantitative scales. In this multi-scale setting it is typically an extremely difficult task to perform approximations, such as Langevin approximations or law of large number type arguments, to simplify a system. Therefore, numerical methods oftentimes are the only reasonable means by which such models can be understood in real time. In this paper we provide a general framework for understanding the weak error of numerical approximation techniques in the multi-scale setting . We quantify how the error of three different methods depends upon both the natural scalings within a given system, and with the step-size of the numerical method. Further, we introduce a new algorithm in this setting, the weak trapezoidal algorithm, which was developed originally as an approximate method for diffusion processes , and prove that the leading order of the error process scales with the square of the time discretization, making it the first second order method in this setting .
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In this paper we provide a general framework for understanding the weak error of numerical approximation techniques in this setting.
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scales in that the different species and reaction rates vary over several orders of magnitude. Quantifying how different
temporal and other quantitative scales. In this multi-scale setting it is typically an extremely difficult task to perform approximations, such as Langevin approximations or law of large number type arguments, to simplify a system. Therefore, numerical methods oftentimes are the only reasonable means by which such models can be understood in real time. In this paper we provide a general framework for understanding the weak error of
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behave in this settingtherefore requires that these scalings be taken into account in an appropriate manner
in the multi-scale setting
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three
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We show that Euler's method, also called explicit tau-leaping, acts as an order one method, in that the error decreases linearly with the step-size, and that the approximate midpoint method acts as either an order one or two method, depending on the relation between the time-step and the scalings in the system.
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has been studied previously only in the diffusive setting
was developed originally as an approximate method for diffusion processes
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it is second order accurate in the size
the leading order of the error process scales with the square
R
of its kind. In the multi-scale setting it is typically an extremely difficult task to perform approximations, such as Langevin approximations or law of large number type arguments, to simplify or reduce a system. Therefore, numerical methods oftentimes are the only reasonable means by which such models can be understood in real time
second order method in this setting