Abstract
In this paper, all the terms in the stochastic Magnus expansion are presented by rooted trees. First, stochastic Magnus methods for linear stochastic differential equations are constructed by truncating the stochastic Magnus expansion. Then, explicit stochastic Magnus methods are constructed by Picard’s iteration for nonlinear stochastic differential equations on matrix Lie group. Furthermore, general nonlinear stochastic differential equations are transformed into linear operator stochastic differential equations by using the Lie derivative. Finally, numerical methods for general nonlinear stochastic differential equations are constructed by using the theory of the stochastic Magnus expansion for the linear case. In particular, for the commutative case, it is shown that the stochastic Magnus expansion provides a novel way to construct computationally inexpensive and arbitrarily high-order numerical methods while avoiding the simulation of multiple stochastic integrals. Moreover, the proposed methods are shown to preserve the intrinsic properties of the original system well and the numerical experiments agree with the theoretical results.
Similar content being viewed by others
References
Blanes, S., Casas, F., Oteo, J.A., Ros, J.: The Magnus expansion and some of its applications. Phys. Rep. Rev. Sect. Phys. Lett. 470(5–6), 151 (2009)
Burrage, K., Burrage, P.M.: High strong order methods for non-commutative stochastic ordinary differential equation systems and the Magnus formula. Phys. D Nonlinear Phenom. 133(1–4), 34 (1999)
Burrage, P.M., Burrage, K.: Structure-preserving Runge–Kutta methods for stochastic Hamiltonian equations with additive noise. Numer. Algorithms 65(3), 519 (2014)
Cao, Z., Grima, R.: Linear mapping approximation of gene regulatory networks with stochastic dynamics. Nat. Commun. 9(1), 3305 (2018)
Casas, F., Iserles, A.: Explicit Magnus expansions for nonlinear equations. J. Phys. A Math. Gen. 39(19), 5445 (2006)
Celledoni, E., Owren, B.: Preserving first integrals with symmetric Lie group methods. Discrete Contin. Dyn. Syst. 34(3), 977 (2014)
Celledoni, E., Marthinsen, H., Owren, B.: An introduction to Lie group integrators-basics, new developments and applications. J. Comput. Phys. 257(2), 1040 (2014)
Crouch, P.E., Grossman, R.: Numerical integration of ordinary differential equations on manifolds. J. Nonlinear Sci. 3(1), 1 (1993)
Debrabant, K., Kværnø, A.: B-series analysis of stochastic Runge–Kutta methods that use an iterative scheme to compute their internal stage values. SIAM J. Numer. Anal. 47(1), 181 (2008)
Debrabant, K., Kværnø, A.: Cheap arbitrary high order methods for single integrand SDEs. BIT Numer. Math. 57(1), 153 (2017)
Hairer, E., Hochbruck, M., Iserles, A., Lubich, C.: Geometric Numerical Integration Geometric Numerical Integration. Springer, Berlin (2002)
Iserles, A., Macnamara, S.: Applications of Magnus expansions and pseudospectra to Markov processes. Eur. J. Appl. Math. 30(2), 400 (2019)
Iserles, A., Nørsett, S.P.: On the solution of linear differential equations in Lie groups. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 357(1754), 983 (1999)
Iserles, A., Munthe-Kaas, H.Z., Nørsett, S.P., Zanna, A.: Lie-group methods. Acta Numer. 9(2), 215 (2000)
Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1992)
Li, X., Zhang, C., Ma, Q., Ding, X.: Discrete gradient methods and linear projection methods for preserving a conserved quantity of stochastic differential equations. Int. J. Comput. Math. 95(12), 2511 (2018)
Magnus, W.: On the exponential solution of differential equations for a linear operator. Commun. Pure Appl. Math. 7, 649 (1954)
Malham, S.J.A., Wiese, A.: Stochastic Lie group integrators. SIAM J. Sci. Comput. 30(2), 597 (2008)
Mananga, E.S., Thibault, C.: Introduction of the Floquet–Magnus expansion in solid-state nuclear magnetic resonance spectroscopy. J. Chem. Phys. 135(4), 044109 (2011)
Mao, X.: Stochastic Differential Equations and Applications Stochastic Differential Equations and Applications. Horwood Publishing, Chichester (2007)
Milstein, G.N.: Numerical Integration of Stochastic Differential Equations Numerical Integration of Stochastic Differential Equations. Kluwer Academic Publishers, Dordrecht (1995)
Munthe-Kaas, H.Z.: Runge–Kutta methods on Lie groups. BIT Numer. Math. 38(1), 92 (1998)
Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1993)
Owren, B.: Order conditions for commutator-free Lie group methods. J. Phys. A Math. Gen. 39(19), 5585 (2006)
Quispel, G.R.W., Mclaren, D.I.: A new class of energy-preserving numerical integration methods. J. Phys. A Math. Theor. 41(4), 045206 (2008)
Tang, Y., Cao, J., Liu, X., Sun, Y.: Symplectic methods for the Ablowitz–Ladik discrete nonlinear Schrödinger equation. J. Phys. A Math. Theor. 40(10), 2425 (2007)
Acknowledgements
We would like to express our sincere thanks to the editor and the reviewers for their careful review and valuable comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Dr. Paul Newton.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is supported by the National Key R&D Program of China (No. 2017YFC1405600), the National Natural Science Foundation of China (Nos. 11501150 and 11701124), and the Natural Science Foundation of Shandong Province of China (No. ZR2017PA006).
Rights and permissions
About this article
Cite this article
Wang, Z., Ma, Q., Yao, Z. et al. The Magnus Expansion for Stochastic Differential Equations. J Nonlinear Sci 30, 419–447 (2020). https://doi.org/10.1007/s00332-019-09578-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00332-019-09578-9