Geodesic
Classification of explicit three-stage symplectic difference schemes for the numerical solution of natural Hamiltonian systems: A comparative study of the accuracy of high-order schemes on molecular dynamics problems
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 4, pp. 551-571

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The natural Hamiltonian systems (systems with separable Hamiltonians) are considered. The variety of explicit three-stage symplectic schemes is described. A classification of the third-order accurate schemes is given. All fourth-order schemes are found (there are seven of them). It is proved that there are no fifth-order schemes. The schemes with improved properties, such as invertibility and optimality with respect to the phase error, are listed. Numerical results that demonstrate the properties of these schemes are presented, and their comparative analysis with respect to the accuracy-efficiency criterion is given. The disbalance of total energy is used as the accuracy criterion. 
@article{ZVMMF_2016_56_4_a3,
     author = {V. N. Sofronov and V. E. Shemarulin},
     title = {Classification of explicit three-stage symplectic difference schemes for the numerical solution of natural {Hamiltonian} systems: {A} comparative study of the accuracy of high-order schemes on molecular dynamics problems},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {551--571},
     publisher = {mathdoc},
     volume = {56},
     number = {4},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_4_a3/}
}
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V. N. Sofronov; V. E. Shemarulin. Classification of explicit three-stage symplectic difference schemes for the numerical solution of natural Hamiltonian systems: A comparative study of the accuracy of high-order schemes on molecular dynamics problems. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 4, pp. 551-571. http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_4_a3/