Abstract
Variational integrators are derived for structure-preserving simulation of stochastic forced Hamiltonian systems. The derivation is based on a stochastic discrete Hamiltonian, which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and considering a stochastic generalization of the deterministic Lagrange–d’Alembert principle. Our approach presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators satisfy a discrete version of the stochastic Lagrange–d’Alembert principle, and in the presence of symmetries, they also satisfy a discrete counterpart of Noether’s theorem. Furthermore, mean-square and weak Lagrange–d’Alembert Runge–Kutta methods are proposed and tested numerically to demonstrate their superior long-time numerical stability and energy behaviour compared to nongeometric methods. The Vlasov–Fokker–Planck equation is considered as one of the numerical test cases, and a new geometric approach to collisional kinetic plasmas is presented.
Original language | English |
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Pages (from-to) | 1318–1367 |
Number of pages | 50 |
Journal | IMA Journal of Numerical Analysis |
Volume | 41 |
Issue number | 2 |
Early online date | 22 Jul 2020 |
DOIs | |
Publication status | Published - Apr 2021 |
Externally published | Yes |
Keywords
- n/a OA procedure
- stochastic forced Hamiltonian systems
- geometric integration
- stochastic variational integrators
- collisional plasma kinetics