Abstract
The paper deals with numerical discretizations of separable nonlinear Hamiltonian systems with additive noise. For such problems, the expected value of the total energy, along the exact solution, drifts linearly with time. We present and analyze a time integrator having the same property for all times. Furthermore, strong and weak convergence of the numerical scheme along with efficient multilevel Monte Carlo estimators are studied. Finally, extensive numerical experiments illustrate the performance of the proposed numerical scheme.
Highlights
Hamiltonian systems are universally used as mathematical models to describe the dynamical evolution of physical systems in science and engineering
The recent years have seen a lot of research activities in the design and numerical analysis of energy-preserving numerical integrators for deterministic Hamiltonian systems, see for instance [3, 4, 7, 13, 21, 22, 29, 31, 33, 35,36,37, 41] and references therein
This is due to the fact that most of the geometrical features underlying deterministic Hamiltonian mechanics are preserved in the Stratonovich setting
Summary
Hamiltonian systems are universally used as mathematical models to describe the dynamical evolution of physical systems in science and engineering. One can show that the expected value of the Hamiltonian (along the exact trajectories) drifts linearly with time, leading to a so-called trace formula, see for instance the work [40] in the case of a linear stochastic oscillator. Proper stochastic perturbations of symplectic Runge–Kutta methods have been investigated as drift-preserving schemes These methods seem effective only for linear problems, while, in the nonlinear case, the drift is not accurately preserved in time. In the case of additive noise, the more the amplitude of the stochastic term increases, the more the accuracy of the drift preservation deteriorates (see, for instance, Table 1 in [5]) This evidence reveals a gap in the existing literature that requires ad hoc numerical methods, effective in preserving the drift in the expected Hamiltonian for the nonlinear case.
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