A Numerical Scheme for Harmonic Stochastic Oscillators Based on Asymptotic Expansions

In this work, we provide a numerical method for discretizing linear stochastic oscillators with high constant frequencies driven by a nonlinear time-varying force and a random force. The presented method is constructed by starting from the variation of constants formula, in which highly oscillating integrals appear. To provide a suited discretisation of this type of integrals, we propose quadrature rules based on asymptotic expansions. Theoretical considerations and numerical experiments comparing the method with a standard approach on physical models are introduced.