Integration of the stochastic underdamped harmonic oscillator by the θ-method

Abstract

[EN]In recent papers, a simple harmonic oscillator with additive noise has been studied by several researchers, and it has been shown that its mean total energy increases linearly as time goes to infinity. In contrast to them, we consider an underdamped harmonic oscillator with additive noise. Our analysis reveals that the mean total energy of the stochastic underdamped harmonic oscillator remains bounded and it asymptotically tends to a certain value. In addition, we give a relation between the mean kinetic energy and the growth rate of the mean total energy. Whereas all stochastic -methods preserve this relation as they are of weak second local order, we show that only the stochastic trapezoidal method can attain the asymptotic values of the mean total energy and its derivative given by the exact solution. Numerical experiments are carried out to confirm these results.