Abstract
We propose explicit stochastic Runge–Kutta (RK) methods for high-dimensional Itô stochastic differential equations. By providing a linear error analysis and utilizing a Strang splitting-type approach, we construct them on the basis of orthogonal Runge–Kutta–Chebyshev methods of order 2. Our methods are of weak order 2 and have high computational accuracy for relatively large time-step size, as well as good stability properties. In addition, we take stochastic exponential RK methods of weak order 2 as competitors, and deal with implementation issues on Krylov subspace projection techniques for them. We carry out numerical experiments on a variety of linear and nonlinear problems to check the computational performance of the methods. As a result, it is shown that the proposed methods can be very effective on high-dimensional problems whose drift term has eigenvalues lying near the negative real axis and whose diffusion term does not have very large noise.
Similar content being viewed by others
Data Availability
Data and program codes are available in https://github.com/yosh-komori/supplementary_info_files_2023.
Notes
For an implementation of the methods, we utilize the parameter values in a Fortran code, rectp.f, obtained from http://anmc.epfl.ch/Pdf/srock2.zip.
References
Abdulle, A., Cirilli, S.: S-ROCK: Chebyshev methods for stiff stochastic differential equations. SIAM J. Sci. Comput. 30(2), 997–1014 (2008)
Abdulle, A., Li, T.: S-ROCK methods for stiff Itô SDEs. Commun. Math. Sci. 6(4), 845–868 (2008)
Abdulle, A., Medovikov, A.A.: Second order Chebyshev methods based on orthogonal polynomials. Numer. Math. 90, 1–18 (2001)
Abdulle, A., Vilmart, G., Zygalakis, K.C.: Weak second order explicit stabilized methods for stiff stochastic differential equations. SIAM J. Sci. Comput. 35(4), A1792–A1814 (2013)
Abdulle, A., Almuslimani, I., Vilmart, G.: Optimal explicit stabilized integrator of weak order 1 for stiff and ergodic stochastic differential equations. SIAM/ASA J. Uncertain. Quantif. 6(2), 937–964 (2018)
Al-Mohy, A.H., Higham, N.J.: A new scaling and squaring algorithm for the matrix exponential. SIAM J. Matrix Anal. 31(3), 970–989 (2009)
Arnold, L.: Stochastic Differential Equations: Theory and Applications. John Wiley & Sons, New York (1974)
Brzeźniak, Z., Goldys, B., Neklyudov, M.: Multidimensional stochastic Burgers equation. SIAM J. Math. Anal. 46(1), 871–889 (2014)
Cohen, D.: On the numerical discretisation of stochastic oscillators. Math. Comput. Simul. 82(8), 1478–1495 (2012)
Cohen, D., Sigg, M.: Convergence analysis of trigonometric methods for stiff second-order stochastic differential equations. Numer. Math. 121(1), 1–29 (2012)
Debrabant, K., Kværnø, A., Mattsson, N.C.: Runge-Kutta Lawson schemes for stochastic differential equations. BIT Numer. Math. 61(2), 381–409 (2021)
Debrabant, K., Kværnø, A., Mattsson, N.C.: Lawson schemes for highly oscillatory stochastic differential equations and conservation of invariants. BIT Numer. Math. 62(4), 1121–1147 (2022)
Dyöngy, I.: Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise I. Potential Anal. 9, 1–25 (1998)
Erdoğan, U., Lord, G.J.: A new class of exponential integrators for SDEs with multiplicative noise. IMA J. Appl. Math. 39(2), 820–846 (2019)
Hochbruck, M., Ostermann, A.: Explicit exponential Runge-Kutta methods for semilinear parabolic problems. SIAM J. Numer. Anal. 43(3), 1069–1090 (2005)
Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209–286 (2010)
Hochbruck, M., Lubich, C., Selhofer, H.: Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput. 19(5), 1552–1574 (1998)
Ibáñez, J., Alonso, J.M., Alonso-Jordá, P., Defez, E., Sastre, J.: Two Taylor algorithms for computing the action of the matrix exponential on a vector. Algorithms 15(2), 48 (2022)
Kamm, K., Pagliarani, S., Pascucci, A.: Numerical solution of kinetic SPDEs via stochastic Magnus expansion. Math. Comput. Simul. 207, 189–208 (2023)
Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer-Verlag, Berlin (1999). (Corrected Third Printing)
Komori, Y., Burrage, K.: Weak second order S-ROCK methods for Stratonovich stochastic differential equations. J. Comput. Appl. Math. 236(11), 2895–2908 (2012)
Komori, Y., Burrage, K.: A stochastic exponential Euler scheme for simulation of stiff biochemical reaction systems. BIT 54(4), 1067–1085 (2014)
Komori, Y., Cohen, D., Burrage, K.: Weak second order explicit exponential Runge-Kutta methods for stochastic differential equations. SIAM J. Sci. Comput. 39(6), A2857–A2878 (2017)
Matsumoto, M., Nishimura, T.: Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator. ACM Trans. Model. Comput. Simul. 8(1), 3–30 (1998)
Milstein, G.N.: Weak approximation of solutions of systems of stochastic differential equations. Theory Probab. Appl. 30(4), 750–766 (1986)
Milstein, G.N., Tretyakov, M.V.: Stochastic Numerics for Mathematical Physics. Springer-Verlag, Berlin (2004)
Moler, C., Loan, C.V.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45(1), 3–49 (2003)
Sidje, R.B.: Expokit: a software package for computing matrix exponentials. ACM Trans. Math. Softw. 24(1), 130–156 (1998)
Tocino, A., Senosiain, M.J.: Mean-square stability analysis of numerical schemes for stochastic differential systems. J. Comput. Appl. Math. 236(10), 2660–2672 (2012)
van der Houwen, P.J., Sommeijer, B.P.: On the internal stability of explicit, \(m\)-stage Runge-Kutta methods for large \(m\)-values. ZAMM Z. Angew. Math. Mech. 60, 479–485 (1980)
Yang, G., Burrage, K., Burrage, P., Ding, X.: How to choose an appropriate initial condition to simulate stochastic differential equations stably. In: AIP Conference Proceedings (to appear)
Yang, G., Burrage, K., Komori, Y., Burrage, P.M., Ding, X.: A class of new Magnus-type methods for semi-linear non-commutative Itô stochastic differential equations. Numer. Algorithms 88(4), 1641–1665 (2021)
Acknowledgements
The authors would like to thank referees, Professor Chi-Wang Shu and Professor David Cohen for their comments which helped to improve the earlier versions of this paper.
Funding
This work was partially supported by JSPS Grant-in-Aid for Scientific Research 17K05369.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have not disclosed any competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Komori, Y., Burrage, K. Split S-ROCK Methods for High-Dimensional Stochastic Differential Equations. J Sci Comput 97, 62 (2023). https://doi.org/10.1007/s10915-023-02354-8
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10915-023-02354-8
Keywords
- Explicit method
- Weak second order approximation
- Orthogonal Runge–Kutta–Chebyshev method
- Stiffness
- Noncommutative noise
- Itô stochastic differential equation