The explicit approximation approach to solve stiff chemical Langevin equations
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The explicit approximation approach to solve stiff chemical Langevin equations Kazem Nouria
, Hassan Ranjbar, Leila Torkzadeh
Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, P. O. Box 35195-363, Semnan, Iran Received: 15 June 2020 / Accepted: 7 September 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The chemical Langevin equations are reputable simulation schemes to explore the dynamics of chemical systems. We propose a new approach to simulate stochastic equations in stiff chemical reactions. The solution procedure is based on the Euler–Maruyama scheme and exponential term. The efficiency and accuracy of our method are studied by two numerical implementations.
1 Introduction The stochastic simulations of different phenomena and chemical kinetics have been studied in the recent years, increasingly [1–13]. For describing the uncertain behaviors of chemical reaction systems, many approaches have been proposed, such as the chemical master equation (CME) [14], chemical Fokker–Plank equation (CFPE) [15], modified Schlögl model [16], and chemical Langevin equations (CLEs) [17]. It is a remarkable fact that molecules show degrees of randomness in their dynamic behavior. We know that the internal noise in random chemical reaction systems characterized by the CME is almost equivalent to the reverse square root of average number of molecules, when reaction species are in low concentrations [8]. Owing to consideration of internal noise, the dynamical behavior of chemical systems described by the CME is more valid than its deterministic models [18]. Explicit closed-form solutions of the CMEs and CLEs are usually not available, and generally it has to be solved numerically. [19,20] established an implicit Milstein method and simulated system of CLEs. Reshniak et al. [21] suggested split-step Milstein schemes to solve the CLEs. Adaptive time-stepping for solution of CLEs was considered by Ilie and Teslya [22], Sotiropoulos and Kaznessis [23]. In this work, we present numerical solution approach by adding exponential term to the Euler–Maruyama scheme. To indicate the efficacy of our scheme, we simulate CLEs which is an Itô stochastic differential equation (SDE) accompanied by a multidimensional Wiener process. The content of this study is as follows. The next section gives the chemical Langevin equations. We present the introduction of stochastic numerical methods in Sect. 3. Numerical illustrations are provided in Sect. 4, and accuracy of the proposed approach is also shown. Finally, a brief conclusion is given in Sect. 5.
a e-mail: [email protected] (corresponding author)
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2 Chemical Langevin systems We introduce the CLEs very briefly. For derivations and further details, we refer the reader to [9,17,22,24–28]. Consider a biochemical system that is well stirred in a constant volume Ω and it is in thermal equilibrium at certain constant temperatures. Le
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