A chemical reaction network is a chemical system involving multiple reactions and chemical species. The simpleststochastic 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 methodsoftentimes 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  .
The simplest, and most common, stochastic model for population processes, including those from biochemistry and cell biology, are continuous time Markov chains. Simulation of such models is often relatively straightforward as there are easily implementable methods for the generation of exact sample paths. However, when using ensemble averages to approximate expected values, the computational complexity can become prohibitive as the number of computations per path scales linearly with the number of jumps, or reactions, of the process. When such methods become computationally intractable, approximate methods, which introduce a bias, can become advantageous . In this paper , we provide a general framework for understanding the weak error , or bias, induced by different numerical approximation techniques . The analysis takes into account both the natural scalings within a given system and the step-size of the numerical method. Further, the weak trapezoidal method is introduced in the current setting, and is proven to be second order accurate in a weak sense , making it the first higher order method in this setting . Examples are provided demonstrating both the main analytical results, and the reduction in computational complexity achieved with the approximate methods .
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A chemical reaction network is a chemical system involving multiple reactions and chemical species. The simpleststochastic models of such networks treat the system as a
The simplest, and most common, stochastic model for population processes, including those from biochemistry and cell biology, are
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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 methodsoftentimes are the only reasonable means by which such models can be understood in real time
chains. Simulation of such models is often relatively straightforward as there are easily implementable methods for the generation of exact sample paths. However, when using ensemble averages to approximate expected values, the computational complexity can become prohibitive as the number of computations per path scales linearly with the number of jumps, or reactions, of the process. When such methods become computationally intractable, approximate methods, which introduce a bias, can become advantageous
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in the multi-scale setting. We quantify how the error of three different methods depends upon
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we introduce a new algorithm in this setting,
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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
method is introduced in the current setting, and is proven to be second order accurate in a weak sense
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. Examples are provided demonstrating both the main analytical results, and the reduction in computational complexity achieved with the approximate methods