Drift-preserving numerical integrators for stochastic Hamiltonian systems

• Published in: Advances in computational mathematics Vol. 46; no. 2
• Main Authors: Chen, Chuchu, Cohen, David, D’Ambrosio, Raffaele, Lang, Annika
• Format: Journal Article
• Language: English
• Published: New York Springer US 2020; Springer Nature B.V
• Subjects: Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; convergence; differential-equations; discretization; driven; Energy; Exact solutions; Hamiltonian functions; Integrators; invariants; Matematik; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematical sciences; Mathematics; Mathematics and Statistics; Multilevel Monte Carlo; Nonlinear systems; Numerical schemes; runge-kutta methods; simulation; stage; Stochastic differential equations; Stochastic Hamiltonian systems; Strong convergence; Trace formula; Visualization; wave-equations; Weak; Weak convergence; Weak convergence; 65C20; Stochastic differential equations; Strong convergence; 60H10; 65C30; Energy; Trace formula; Multilevel Monte Carlo; Numerical schemes; Stochastic Hamiltonian systems; 60H35
• ISSN: 1019-7168; 1572-9044; 1572-9044
• DOI: 10.1007/s10444-020-09771-5

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.