In this work, we propose a Lie algebraic approach for numerically solving a class of highly oscillatory stochastic Hamiltonian systems (SHSs). For a concrete highly oscillatory SHS, we construct two numerical schemes based on the Lie algebraic approach, and prove their near preservation of the symplecticity. We also show by numerical tests their root mean-square convergence orders, as well as their effectiveness and merits in solving the highly oscillatory SHS.
RUAN Jialin
,
WANG Lijin
. A Lie algebraic approach for a class of highly oscillatory stochastic Hamiltonian systems[J]. Journal of University of Chinese Academy of Sciences, 2019
, 36(1)
: 5
-10
.
DOI: 10.7523/j.issn.2095-6134.2019.01.002
[1] Higham D J. An algorithmic introduction to numerical simulation of stochastic differential equations[J]. SIAM review, 2001, 43(3):525-546.
[2] Misawa T. A Lie algebraic approach to numerical integration of stochastic differential equations[J]. SIAM Journal on Scientific Computing, 2001, 23(3):866-890.
[3] Feng K, Qin M. Symplectic geometric algorithms for hamiltonian systems[M]. Berlin:Springer, 2010.
[4] Milstein G N, Repin Y M, Tretyakov M V. Symplectic integration of Hamiltonian systems with additive noise[J]. SIAM Journal on Numerical Analysis, 2002,39(6):2066-2088.
[5] Milstein G N, Repin Y M, Tretyakov M V. Numerical methods for stochastic systems preserving symplectic structure[J]. SIAM Journal on Numerical Analysis, 2002, 40(4):1583-1604.
[6] Kloeden P E, Platen E. Numerical solution of stochastic differential equations[M]. Springer-Verlag, 1992.
[7] Milstein G N. Numerical integration of stochastic differential equations[M]. Springer Science & Business Media, 1994.
[8] Milstein G N, Tretyakov M V. Stochastic numerics for mathematical physics[M]. Springer Science & Business Media, 2013.
[9] Chartier P, Makazaga J, Murua A, et al. Multi-revolution composition methods for highly oscillatory differential equations[J]. Numerische Mathematik, 2014, 128(1):167-192.
[10] Vilmart G. Weak second order multirevolution composition methods for highly oscillatory stochastic differential equations with additive or multiplicative noise[J]. Siam Journal on Scientific Computing, 2015, 36(4):A1770-A1796.
[11] Cohen D. On the numerical discretisation of stochastic oscillators[J]. Mathematics & Computers in Simulation, 2012, 82(8):1478-1495.
[12] Cohen D. Convergence analysis of trigonometric methods for stiff second-order stochastic differential equations[J]. Numerische Mathematik, 2012, 121(1):1-29.
[13] Hairer E, Lubich C, Wanner G. Geometric numerical integration:structure-preserving algorithms for ordinary differential equations[J]. Series in Computational Mathematics, 2006, 25(1):805-882.
[14] Kunita H. On the representation of solutions of stochastic differential equations[M]. Séminaire de Probabilités XIV 1978/79. Springer Berlin Heidelberg, 1980:282-304.
