Stochastic discrete Hamiltonian variational integrators.

Area of Science:

• Geometric mechanics
• Computational physics
• Numerical analysis

Background:

• Stochastic Hamiltonian systems are crucial in fields like geometric mechanics.
• Existing numerical methods often struggle with long-term stability and energy conservation for these systems.
• Multiplicative noise poses particular challenges for traditional integrators.

Purpose of the Study:

• To derive novel variational integrators for structure-preserving simulation of stochastic Hamiltonian systems.
• To develop a unified framework for existing and new structure-preserving numerical schemes.
• To enhance the long-time numerical stability and energy behavior of simulations.

Main Methods:

• Approximation of a type-II stochastic generating function using a stochastic discrete Hamiltonian.
• Application of a variational principle to a stochastic action functional.
• Development of a general methodology for deriving new structure-preserving numerical schemes.

Main Results:

• The derived integrators are symplectic and preserve integrals of motion related to Lie group symmetries.
• Stochastic symplectic Runge-Kutta methods are a special case of the developed integrators.
• New low-stage stochastic symplectic methods of mean-square order 1.0 were developed and numerically tested.

Conclusions:

• The new variational integrators demonstrate superior long-time numerical stability and energy behavior compared to nonsymplectic methods.
• The unified framework simplifies the study and development of structure-preserving integrators.
• This approach provides a powerful tool for accurate and stable simulations in geometric mechanics.