Advertisement

Iterative Semi-implicit Splitting Methods for Stochastic Chemical Kinetics

  • Jürgen GeiserEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11386)

Abstract

In this paper, we present splitting methods that are based on iterative schemes and applied to stochastic models for chemical kinetics. The motivation arose of solving chemical kinetics with respect to stochastic influences in their models. The parameters and variables that describe the concentrations are based on extending the deterministic models to stochastic models. Such an extension is important to simulate the uncertainties of the concentrations. For the modelling equations, we deal with stochastic differential equations and it is important to extend the deterministic methods to stochastic methods. Here, we consider iterative splitting methods, based on Picard’s successive approximations, to solve the underlying stochastic differential equations. The benefit of relaxation behaviour of the iterative solvers is also obtained in the stochastic method and based on the stochastic process we obtain 1/2 of the accuracy as for the deterministic method, which is also given for non-iterative methods. We present the numerical analysis of the schemes and verified the results in numerical experiments of different chemical reaction systems.

Keywords

Splitting methods Stochastic differential equations Iterative splitting schemes Convergence analysis Chemical reaction systems 

References

  1. 1.
    Burkholder, D.L., Davis, B., Gundy, R.F.: Integral inequalities for convex functions of operators on martingales. In: Proceedings of 6th Berkeley Symposium on Mathematical Statistics and Probability, vol. 2, pp. 223–240 (1972)Google Scholar
  2. 2.
    Cao, Y., Li, H., Petzold, L.: Efficient formulation of the stochastic simulation algorithm for chemically reacting systems. J. Chem. Phys. 121(9), 4059–4067 (2004)CrossRefGoogle Scholar
  3. 3.
    Fan, Z.: SOR waveform relaxation methods for stochastic differential equations. Appl. Math. Comput. 219, 4992–5003 (2013)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Geiser, J.: Iterative Splitting Methods for Differential Equations, Numerical Analysis and Scientific Computing Series. CRC Press, Chapman & Hall/CRC, Boca Raton (2011). Edited by Magoules and LaiCrossRefGoogle Scholar
  5. 5.
    Geiser, J.: Computing exponential for iterative splitting methods. J. Appl. Math. 2011, Article ID 193781 (2011)Google Scholar
  6. 6.
    Geiser, J.: Multiscale splitting for stochastic differential equations: applications in particle collisions. J. Coupled Syst. Multiscale Dyn. 1, 241–250 (2013)CrossRefGoogle Scholar
  7. 7.
    Gillespie, D.T.: Approximate accelerated stochastic simulation of chemically reacting systems. J. Chem. Phys. 115(4), 1716–1733 (2001)CrossRefGoogle Scholar
  8. 8.
    Izzo, A.: \(C^r\) convergence of Picard’s sucessive approximations. Proc. Am. Math. Soc. 127(7), 2059–2063 (1999)CrossRefGoogle Scholar
  9. 9.
    Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations. SIAM Frontiers in Applied Mathematics, vol. 16. SIAM, Philadelphia (1995)CrossRefGoogle Scholar
  10. 10.
    Kloeden, P.E., Platen, E.: The Numerical Solution of Stochastic Differential Equations. Springer, Heidelberg (1992).  https://doi.org/10.1007/978-3-662-12616-5CrossRefzbMATHGoogle Scholar
  11. 11.
    Kafash, B., Lalehzari, R., Delavarkhalafi, A., Mahmoudi, E.: Application of stochastic differential system in chemical reactions via simulation. MATCH Commun. Math. Comput. Chem. 71, 265–277 (2014)Google Scholar
  12. 12.
    Li, Z., Liu, J.: \(C^{\infty }\)-convergence of Picard’s successive approximations to solutions of stochastic differential equations. Stat. Probab. Lett. 129, 203–209 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Miekkala, U., Nevanlinna, O.: Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Oksendal, B.: Stochastic Differential Equations. Springer, Berlin (2003).  https://doi.org/10.1007/978-3-642-14394-6CrossRefzbMATHGoogle Scholar
  15. 15.
    Silva-Dias, L., Lopez-Castillo, A.: Practical stochastic model for chemical kintetics. Quim. Nova 38(9), 1232–1236 (2015)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Electrical Engineering and Information TechnologyRuhr University of BochumBochumGermany

Personalised recommendations