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Modified Verlet Method Involving Second-Order Mid-Point Rule Applied to Balls Falling in One-Dimensional Potentials

Published online by Cambridge University Press:  28 May 2015

Hidenori Yasuda
Hidenori Yasuda*
Affiliation:
Department of Mathematics, Faculty of Science, Josai University, 1-1 Keyakidai, Sakado, Saitama, Japan
*
Corresponding author. Email: yasuda@math.josai.ac.jp
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Abstract

A modified Verlet method which involves a kind of mid-point rule is constructed and applied to the one-dimensional motion of elastic balls of finite size, falling under constant gravity in space and then under the chemical potential in the interface region of phase separation within a two-liquid film. When applied to the simulation of two balls falling under constant gravity in space, the new method is found to be computationally superior to the usual Verlet method and to Runge–Kutta methods, as it allows a larger time step for comparable accuracy. The main purpose of this paper is to develop an efficient numerical method to simulate balls in the interface region of phase separation within the two-liquid film, where the ball motion is coupled with two-phase flow. The two-phase flow in the film is described via shallow water equations, using an invariant finite difference scheme that accurately resolves the interface region. A larger time step in computing the ball motion, more comparable with the time step in computing the two-phase flow, is a significant advantage. The computational efficiency of the new method in the coupled problem is demonstrated for the case of four elastic balls in the two-liquid film.

Keywords

65L1265M0665P10Verlet methodfalling ballsfirst return mapphase separationshallow water equations
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 1 , Issue 1 , February 2011 , pp. 1 - 19
Copyright
Copyright © Global-Science Press 2011

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References

[1]Butcher, J.C. (2003). Numerical Methods of Ordinary Differential Equations, Wiley.Google Scholar
[2]Chaikin, P.M., and Lubensky, T.C. (1995). Principles of Condensed Matter Physics, Cambridge University Press.Google Scholar
[3]Daly, B.J., and Torrey, M.D. (1984). SOLA-PTS: a transient, three-dimensional algorithm for fluid-thermal mixing and wall heat transfer in complex geometries NUREG/CR-3822; LA-10132-MS,Technical Report of Los Alamos National Laboratory.Google Scholar
[4]Galperin, G. (1978). Elastic collisions of particles on a line, Russian Math. Surveys 33, 119–200.Google Scholar
[5]Gidaspow, D. (1994). Multiphase Flow and Fluidization, Academic Press.Google Scholar
[6]Ishii, M. (1975). Thermo-fluid dynamics theory of two-phase flow, Eyrolles.Google Scholar
[7]LeVeque, R.J. (2002). Finite volume method for hyperbolic problems, Cambridge University Press.Google Scholar
[8]Lin, Y., Skaff, H., Emrick, T., Dinsmore, A.D., and Russell, T. (2003). Nanoparticle assembly and transport at liquid-liquid interfaces. Science 299, 226–229.Google Scholar
[9]Peng, G., Qiu, F., Ginzburg, V.V., Jasnow, D., and Balazs, A.C. (2000). Forming supramolecular networks from nanoscale rods in binary, phase-separating mixtures. Science 288, 1802–1804.CrossRefGoogle ScholarPubMed
[10]Shokin, Y.I. (1983). The Method of Differential Approximation, Springer-Verlag.Google Scholar
[11]Timoshenko, S.P., and Goodier, J.N. (1970). Theory of Elasticity, McGraw Hill.Google Scholar
[12]Yanenko, N.N., and Shokin, Y.L. (1971). On the group classification of the difference schemes for systems of equations of gas dynamics. In Lecture Notes in Physics 8 (Holt, M. Ed.), 3–17, Springer.Google Scholar
[13]Yanenko, N.N., and Shokin, Y.L. (1973). Schemes numerique invariants de groupe pour les equations de la dynamique de gas. In Lecture Notes in Physics 18, Cabannes, H., and Temam, R. Eds., 174–186, Springer.Google Scholar
[14]Yasuda, H. (2009). Two-phase shallow water equations and phase separation in thin immiscible liquid films. Journal of Scientific Computing, in press. (On-line vr. DOI: 10.1007/s10915-009-9280-6)Google Scholar
[15]Whelan, N.D., Goodings, D.A., and Cannizo, J.K. (1990). Two balls in one dimension with gravity. Phys. Rev. A42, 742–754.Google Scholar
[16]Wojtkowski, M. (1990). A system of one-dimensional balls with gravity. Commun. Math. Phys. 126, 507–533.Google Scholar
[17]Wojtkowski, M. (1990). The system of one-dimensional balls in an external field. Commun. Math. Phys. 127, 425–432.Google Scholar
[18]Wyrwa, D., Beyer, N., and Schmid, G. (2002). One-dimensional arrangements of metal nan-oclusters. Nano Letters 2, 419-321.Google Scholar