An analysis of the efficiency of higher-order symplectic schemes by the example of a problem of the collision of a nanoparticle with an obstacle


  • Evgenii V. Vorozhtsov


molecular dynamics
Hamilton equations
collision of a nanoparticle with an obstacle
EAM potentials
multi-component systems
symplectic difference schemes


The work is devoted to the application of the molecular dynamics (MD) method for numerical simulation of high-velocity interaction of solids. The phenomenon of the copper nanoparticle rebound from the aluminum plate has been simulated numerically. For the Verlet scheme and the new symplectic scheme FR50 obtained recently by the author, the intervals of time steps were found such that at the specification of the time step within these intervals, no drift of the total energy occurs. The total energy drift arises in the case of the FR50 scheme at much larger time steps than in the case of the Verlet method. The need in introducing several similarity criteria by analogy with the continuum mechanics has been shown by analyzing the results of the numerical simulation of problems involving the nanoparticle collision with an obstacle. A rapid and simple computer test has been proposed for the rejection of the EAM potentials before starting to solve the main MD problem.





Methods and algorithms of computational mathematics and their applications

Author Biography

Evgenii V. Vorozhtsov


  1. S. P. Kiselev, E. V. Vorozhtsov, and V. M. Fomin, Foundations of Fluid Mechanics with Applications: Problem Solving Using Mathematica (Birkh854user, Boston, 1999).
  2. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 1: textitMechanics (Nauka, Moscow, 1973; Pergamon, Oxford, 1977).
  3. S. K. Godunov, S. P. Kiselev, I. M. Kulikov, and V. I. Mali, Modeling of Shockwave Processes in Elastic-Plastic Materials at Different (Atomic, Meso and Thermodynamic) Structural Levels (Inst. Komp’yut. Issled., Moscow-Izhevsk, 2014) [in Russian].
  4. E. V. Vorozhtsov and S. P. Kiselev, “Explicit Higher-Order Schemes for Molecular Dynamics Problems,” Vychisl. Metody Program. (Numerical Methods and Programming) 22 (2), 87-109 (2021).
    doi 10.26089/NumMet.v22r207
  5. E. V. Vorozhtsov and S. P. Kiselev, “Higher-Order Symplectic Integration Techniques for Molecular Dynamics Problems,” J. Comput. Phys. 452, Article ID 110905 (2022).
    doi 10.1016/
  6. L. Verlet, “Computer ’Experiments’ on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules,” Phys. Rev. 159 (1), 98-103 (1967).
    doi 10.1103/PhysRev.159.98
  7. Yu. B. Suris, “The Canonicity of Mappings Generated by Runge-Kutta Type Methods when Integrating the Systems ddotx =-partial U/partial x,” Zh. Vychisl. Mat. Mat. Fiz. 29 (2), 202-211 (1989) [USSR Comput. Math. Math. Phys. 29 (1), 138-144 (1989)].
    doi 10.1016/0041-5553(89)90058-X
  8. W. W. Adams and P. Loustaunau, An Introduction to Gröbner Bases , Vol. 3: Graduate Studies in Mathematics (Amer. Math. Soc., Providence, 1996).
  9. R. D. Ruth, “A Canonical Integration Technique,” IEEE Trans. Nucl. Sci. NS-30 (4), 2669-2671 (1983).
  10. E. Forest and R. D. Ruth, “Fourth-Order Symplectic Integration,” Physica D 43 (1), 105-117 (1990).
    doi 10.1016/0167-2789(90)90019-L
  11. E. V. Vorozhtsov and S. P. Kiselev, “An Efficient Method of Finding New Symplectic Schemes for Hamiltonian Mechanics Problems with the Aid of Parametric Gröbner Bases,” J. Comput. Phys. 496, Article ID 112601 (2024).
    doi 10.1016/
  12. B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry -- Methods and Applications (Nauka, Moscow, 1986; Springer, New York, 1992).
  13. E. Hairer, S. P. Norsett, and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems (Springer, Berlin, 1987; Mir, Moscow, 1990).
  14. J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems (Chapman and Hall, London, 1994).
  15. B. Leimkuhler and S. Reich, Simulating Hamiltonian Dynamics (Cambridge Univ. Press, Cambridge, 2004).
  16. E. Hairer, G. Wanner, and C. Lubich, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (Springer, Berlin, 2006).
    doi 10.1007/3-540-30666-8
  17. M. I. Baskes and C. F. Melius, “Pair Potentials for fcc Metals,” Phys. Rev. B 20 (8), 3197-3204 (1979).
  18. N. I. Papanicolaou, G. C. Kallinteris, G. A. Evangelakis, and D. A. Papaconstantopoulos, “Second-Moment Interatomic Potential for Aluminum Derived from Total-Energy Calculations and Molecular Dynamics Application,” Comput. Mater. Sci. 17 (2-4), 224-229 (2000).
    doi 10.1016/S0927-0256(00)00028-8
  19. M. A. Karolewski, “Tight-Binding Potentials for Sputtering Simulations with fcc and bcc Metals,” Radiat. Eff. Defects Solids 153 (3), 239-255 (2001).
    doi 10.1080/10420150108211842
  20. A. R. West, Solid State Chemistry and Its Applications (Wiley, New York, 1984).
  21. S. M. Foiles, “Calculation of the Surface Segregation of Ni-Cu Alloys with the Use of the Embedded-Atom Method,” Phys. Rev. B 32 (12), 7685-7693 (1985).
    doi 10.1103/PhysRevB.32.7685
  22. S. M. Foiles, M. I. Baskes, and M. S. Daw, “Embedded-Atom-Method Functions for the fcc Metals Cu, Ag, Au, Ni, Pd, Pt, and Their Alloys,” Phys. Rev. B 33 (12), 7983-7991 (1986).
    doi 10.1103/PhysRevB.33.7983
  23. R. A. Johnson, “Alloy Models with the Embedded-Atom Method,” Phys. Rev. B 39 (17), 12554-12559 (1989).
    doi 10.1103/PhysRevB.39.12554
  24. F. Cleri and V. Rosato, “Tight-Binding Potentials for Transition Metals and Alloys,” Phys. Rev. B 48 (1), 22-33 (1993).
    doi 10.1103/PhysRevB.48.22
  25. R. R. Zope and Y. Mishin, “Interatomic Potentials for Atomistic Simulations of the Ti-Al System,” Phys. Rev. B 68 (2), 024102-1-024102-14 (2003).
    doi 10.1103/PhysRevB.68.024102
  26. A. O. E. Animalu, Intermediate Quantum Theory of Crystalline Solids (Prentice-Hall, Englewood Cliffs, 1977; Mir, Moscow, 1981).
  27. A. E. Carlsson, “Beyond Pair Potentials in Elemental Transition Metals and Semiconductors,” Solid State Phys. 43, 1-91 (1990).
    doi 10.1016/S0081-1947(08)60323-9
  28. M. W. Finnis and J. E. Sinclair, “A Simple Empirical N-body Potential for Transition Metals,” Philos. Mag. A 50 (1), 45-55 (1984).
    doi 10.1080/01418618408244210
  29. Y. Mishin, D. Farkas, M. J. Mehl, and D. A. Papaconstantopoulos, “Interatomic Potentials for Monoatomic Metals from Experimental Data and ab initio Calculations,” Phys. Rev. B 59 (5), 3393-3407 (1999).
    doi 10.1103/PhysRevB.59.3393
  30. M. Griebel, G. Zumbusch, and S. Knapek, Numerical Simulation in Molecular Dynamics: Numerics, Algorithms, Parallelization, Applications (Springer, Berlin, 2007).
    doi 10.1007/978-3-540-68095-6
  31. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics , Vol. 5: Statistical Physics , Part 1 (Nauka, Moscow, 1976; Pergamon, Oxford, 1980).
  32. A. V. Gerasimov, V. N. Barashkov, V. P. Glazyrin, et al., Theoretical and Experimental Investigations of the High-Velocity Interaction of Bodies (Tomsk Gos. Univ., Tomsk, 2007) [in Russian].
  33. A. P. Alkhimov, V. F. Kosarev, and A. N. Papyrin, “A Method of Cold Gas-Dynamic Spraying,” Dokl. Akad. Nauk SSSR 315 (5), 1062-1065 (1990).
  34. A. P. Alkhimov, S. V. Klinkov, V. F. Kosarev, et al., “Heterogeneous Technologies: Particle-Obstacle Interaction Problems,” Teplofiz. Aeromekh. 12 (3), 415-432 (2005) [Thermophys. Aeromech. 12 (3), 387-402 (2005)].
  35. K. Feng, “Difference Schemes for Hamiltonian Formalism and Symplectic Geometry,” J. Comput. Math. 4 (3), 279-289 (1986).
  36. S. P. Kiselev, “Method of Molecular Dynamics in Mechanics of Deformable Solids,” Zh. Prikl. Mekh. Tekh. Fiz. 55 (3), 113-139 (2014) [J. Appl. Mech. Tech. Phys. 55 (3), 470-493 (2014)].
    doi 10.1134/S0021894414030109
  37. S. Wolfram, The Mathematica Book (Wolfram Media, Champaign, 2003).
  38. J. Gans and D. Shalloway, “Shadow Mass and the Relationship between Velocity and Momentum in Symplectic Numerical Integration,” Phys. Rev. E 61 (4) 4587-4592 (2000).
    doi 10.1103/physreve.61.4587
  39. R. D. Engle, R. D. Skeel, and M. Drees, “Monitoring Energy Drift with Shadow Hamiltonians,” J. Comput. Phys. 206 (2), 432-452 (2005).
    doi 10.1016/
  40. LAMMPS Users Manual 31 Mar 2017 version. Sandia National Laboratories. . Cited May 8, 2024.
  41. A. N. Agafonov and A. V. Eremin, The Method of Classical Molecular Dynamics in the Simulation of Physical and Chemical Processes (Samara Gos. Univ., Samara, 2017) [in Russian].
  42. L. Rovigatti, P. Šulc, I. Z. Reguly, and F. Romano, “A Comparison between Parallelization Approaches in Molecular Dynamics Simulations on GPUs,” J. Comput. Chem. 36 (1), 1-8 (2015).
    doi 10.1002/jcc.23763
  43. P. Mathur, H. K. Azad, S. H. V. Sangaraju, and E. Agrawal, “Parallelization of Molecular Dynamics Simulations Using Verlet Algorithm and OpenMP,” in Lecture Notes in Network and Systems (Springer, Singapore, 2024), Vol. 832, pp. 263-274.
    doi 10.1007/978-981-99-8129-8_22
  44. V. O. Podryga and S. V. Polyakov, “Parallel Implementation of Multiscale Approach to the Numerical Study of Gas Microflows,” Vychisl. Metody Program. (Numerical Methods and Programming) 17 (2), 147-165 (2016).
    doi 10.26089/NumMet.v17r214