## THERMAL SCIENCE

International Scientific Journal

### NUMERICAL DISCRETIZATION OF STOCHASTIC OSCILLATORS WITH GENERALIZED NUMERICAL INTEGRATORS

**ABSTRACT**

In this study, we propose a numerical scheme for stochastic oscillators with additive noise obtained by the method of variation of constants formula using generalized numerical integrators. For both of the displacement and the velocity components, we show that the scheme has an order of 3/2 in one step convergence and a first order in overall convergence. Theoretical statements are supported by numerical experiments.

**KEYWORDS**

PAPER SUBMITTED: 2020-06-30

PAPER REVISED: 2020-11-07

PAPER ACCEPTED: 2020-11-13

PUBLISHED ONLINE: 2021-01-24

- Markus L., Weerasinghe A., Stochastic oscillators, Journal of Differential Equations, 71, 1988, 2, pp. 288-314.
- Senosiain M. J., Tocino A., A review on numerical schemes for solving a linear stochastic oscillator, BIT Numerical Mathematics, 55, 2015, 2, pp. 515-529.
- Arathi S., Rajasekar S., Stochastic resonance in a single-well anharmonic oscillator with coexisting attractors, Communications in Nonlinear Science and Numerical Simulation, 19, 2014, 12, pp. 4049-4056.
- Hong J., Scherer R., Wang L., Predictor-corrector methods for a linear stochastic oscillator with additive noise, Mathematical and Computer Modelling, 46, 2007, 5, pp. 738-764.
- De la Cruz H., Jimenez J. C., Zubelli J. P., Locally Linearized methods for the simulation of stochastic oscillators driven by random forces, BIT Numerical Mathematics, 57, 2017, 1, pp. 123-151.
- El-Tawil M. A., Al-Johani A. S., Approximate solution of a mixed nonlinear stochastic oscillator, Computers & Mathematics with Applications, 58, 2009, 11, pp. 2236-2259.
- Gitterman M., Classical harmonic oscillator with multiplicative noise, Physica A: Statistical Mechanics and its Applications, 352, 2005, 2, pp. 309-334.
- Guo S., Er G., The probabilistic solution of stochastic oscillators with even nonlinearity under poisson excitation, Open Physics, 10, 2012, 3, pp. 702-707.
- Grue J., Øksendal B., A stochastic oscillator with time-dependent damping, Stochastic Processes and their Applications, 68, 1997, 1, pp. 113-131.
- Gitterman M., Stochastic oscillator with random mass: New type of Brownian motion, Physica A: Statistical Mechanics and its Applications, 395, 2014, pp. 11-21.
- Cohen D., On the numerical discretisation of stochastic oscillators, Mathematics and Computers in Simulation, 82, 2012, 8, pp. 1478-1495.
- Ashyralyev A., Akat M., An approximation of stochastic hyperbolic equations: case with Wiener process, Mathematıcal Methods in the Applıed Scıences, 36, 2013, 9, pp. 1095-1106.
- Ashyralyev A., Akat M., (2011), An Approximation of Stochastic Hyperbolic Equations, AIP Conference Proceedings, 1389, DOI: 10.1063/1.3636808.