Comparison of Numerical Methods for the Kuba Oscillator

Year 2025, Volume: 14 Issue: 1, 260 - 272, 26.03.2025

Gülşen Orucova Büyüköz , Tuğçem Partal , Prof. Dr. Mustafa Bayram

https://doi.org/10.17798/bitlisfen.1573596

Abstract

In this study, numerical solutions of stochastic differential equation (SDE) systems have been analyzed and three different numerical methods used for solving these systems, the Milstein method, the Simplified Second-Order Taylor Scheme, and the Stochastic Runge-Kutta (SRK) method, have been compared. The Kubo oscillator model has been considered and the stochastic dynamics of this model have been solved using numerical methods. Initially, the general structure of SDEs is introduced, and the theoretical foundations of the methods used for solving these systems are explained.
In the study, the stochastic model of the Kubo oscillator was solved numerically using the Milstein method, the Simplified Second-Order Taylor Scheme, and the SRK method. The results obtained were compared with exact solutions. In the numerical computations, the accuracy of all three methods is analyzed for different discretization counts and the results were supported by graphs and error tables. The comparisons revealed that the Simplified Second-Order Taylor Scheme provided more accurate solutions compared to the Milstein method. It is observed that the Taylor method and the SRK 2-stage method gave close results. Additionally, it was observed that increasing the number of discretizations brought both methods closer to the exact solution.

Keywords

Stochastic differential equation systems, Numerical approximations, Kubo oscillator

Ethical Statement

The study is complied with research and publication ethics.

References

• X. Mao, Stochastic Differential Equations and Applications, 2nd ed. Elsevier, 2007.
• F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, vol. 81, no. 3, pp. 637-654, 1973.
• D. Cohen, “On The Numerical Discretisation of Stochastic Oscillators,” Mathematics and Computers in Simulation, vol. 82, no. 8, pp. 1478-1495, 2012.
• R. Kubo, “Stochastic Liouville equations”, J. Math. Phys. 4, 174–183, 1963.
• B. J. Berne, R. Pecora, Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics, Wiley-Interscience, 1976.
• C. Xupeng & W. Lijin, “Learning stochastic Hamiltonian systems via neural network and numerical quadrature formulae”, Journal of University of Chinese Academy of Sciences, 74.
• D. Segal, “Vibrational relaxation in the Kubo oscillator: Stochastic pumping of heat.”, The Journal of chemical physics 130.13, 2009.
• G. N. Milstein, “Approximate Integration of Stochastic Differential Equations,” Theory of Probability & Its Applications, vol. 19, no. 3, pp. 557-562, 1975.
• P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Berlin, Heidelberg: Springer, 1992.
• E. Allen, Modeling with Itô Stochastic Differential Equations, vol. 22. Springer Science & Business Media, 2007.
• D. J. Higham, “An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations,” SIAM Review, vol. 43, no. 3, pp. 525-546, 2001.
• R. F. Fox, R. Roy, and A. W. Yu, “Tests of Numerical Simulation Algorithms for the Kubo Oscillator,” Journal of Statistical Physics, vol. 47, pp. 477-487, 1987.
• B. Øksendal, Stochastic Differential Equations, Berlin, Heidelberg: Springer, 2003.
• A. Friedman, Stochastic Differential Equations and Applications, Berlin, Heidelberg: Springer, 1975.
• A. Tocino and R. Ardanuy, “Runge–Kutta Methods for Numerical Solution of Stochastic Differential Equations,” Journal of Computational and Applied Mathematics, vol. 138, no. 2, pp. 219-241, 2002.
• Baylan, E., (2014). Simulating Stochastic Differential Equations Using Ito-Taylor Schemes, Master’s Thesis, METU, Ankara. 3(1):37-51.