Procedure for constructing of explicit, implicit and symmetric simplectic schemes for numerical solving of Hamiltonian systems of equations

Batgerel B., Nikonov E.G., Puzynin I.V.
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Equations of motion in Newtonian and Hamiltonian forms are used for classical molecular dynamics simulation of particle system time evolution. When Newton equations of motion are used for finding of particle coordinates and velocities in $N$-particle system it takes to solve $3N$ ordinary differential equations of second order at every time step. Traditionally numerical schemes of Verlet method are used for solving Newtonian equations of motion of molecular dynamics. A step of integration is necessary to decrease for Verlet numerical schemes steadiness conservation on sufficiently large time intervals. It leads to a significant increase of the volume of calculations. Numerical schemes of Verlet method with Hamiltonian conservation control (the energy of the system) at every time moment are used in the most software packages of molecular dynamics for numerical integration of equations of motion. It can be used two complement each other approaches to decrease of computational time in molecular dynamics calculations. The first of these approaches is based on enhancement and software optimization of existing software packages of molecular dynamics by using of vectorization, parallelization and special processor construction. The second one is based on the elaboration of efficient methods for numerical integration for equations of motion. A procedure for constructing of explicit, implicit and symmetric symplectic numerical schemes with given approximation accuracy in relation to integration step for solving of molecular dynamic equations of motion in Hamiltonian form is proposed in this work. The approach for construction of proposed in this work procedure is based on the following points: Hamiltonian formulation of equations of motion; usage of Taylor expansion of exact solution; usage of generating functions, for geometrical properties of exact solution conservation, in derivation of numerical schemes. Numerical experiments show that obtained in this work symmetric symplectic third-order accuracy scheme conserves basic properties of the exact solution in the approximate solution. It is more stable for approximation step and conserves Hamiltonian of the system with more accuracy at a large integration interval then second order Verlet numerical schemes.

Keywords: Hamiltonian systems of equations, simplectic difference schemes, generating functions, molecular dynamics
Citation in English: Batgerel B., Nikonov E.G., Puzynin I.V. Procedure for constructing of explicit, implicit and symmetric simplectic schemes for numerical solving of Hamiltonian systems of equations // Computer Research and Modeling, 2016, vol. 8, no. 6, pp. 861-871
Citation in English: Batgerel B., Nikonov E.G., Puzynin I.V. Procedure for constructing of explicit, implicit and symmetric simplectic schemes for numerical solving of Hamiltonian systems of equations // Computer Research and Modeling, 2016, vol. 8, no. 6, pp. 861-871
DOI: 10.20537/2076-7633-2016-8-6-861-871
URL: http://crm-en.ics.org.ru/journal/article/2519/
Creative Commons License This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License.
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