Geodesic
New two-level leapfrog scheme for modeling the stochastic Landau–Lifshitz equations
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 54 (2014) no. 2, pp. 298-317 
V. Yu. Glotov; V. M. Goloviznin; S. A. Karabasov; A. P. Markeshteijn. New two-level leapfrog scheme for modeling the stochastic Landau–Lifshitz equations. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 54 (2014) no. 2, pp. 298-317. http://geodesic.mathdoc.fr/item/ZVMMF_2014_54_2_a8/
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     author = {V. Yu. Glotov and V. M. Goloviznin and S. A. Karabasov and A. P. Markeshteijn},
     title = {New two-level leapfrog scheme for modeling the stochastic {Landau{\textendash}Lifshitz} equations},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {298--317},
     year = {2014},
     volume = {54},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2014_54_2_a8/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

A two-level modification of the classical nondissipative leapfrog scheme with nonlinear flux correction has been developed for fluctuating hydrodynamics problems. The new algorithm has shown to satisfy the fluctuation-dissipation theorem to high accuracy. The results of various numerical tests, including equilibrium, nonequilibrium, one-, and two-dimensional systems, are compared with theoretical predictions, direct molecular simulations, and results produced by MacCormack’s schemes, the piecewise parabolic method, and a third-order Runge–Kutta scheme. The new algorithm is well suited for parallel computations due to its implementation simplicity and compact stencil.

[1] Lifshits E. M., Pitaevskii L. P., Statisticheskaya fizika, v. 2, Teoriya kondensirovannogo sostoyaniya, Nauka, M., 1978 | MR

[2] Landau L. D., Lifshits E. M., Statisticheskaya fizika, v. 1, Fizmatlit, M., 2005

[3] Bell J. B., Garcia A. I., Williams S. A., “Numerical methods for the stochastic Landau–Lifshitz Navier–Stokes equations”, Phys. Rev. E, 76:1 (2007), 016708 | DOI | MR

[4] Bell J. B., Garcia A. L., Williams S. A., “Computational fluctuating fluid dynamics”, ESAIM: Math. Modeling and Numer. Anal., 44 (2010), 1085–1105 | DOI | MR | Zbl

[5] Donev A. et al., “On the accuracy of finite-volume schemes for fluctuating hydrodynamics”, Comm. Appl. Math. and Comp. Sci., 5:2 (2010), 149–197 | DOI | MR | Zbl

[6] Voulgarakis N. K., Chu J.-W., “Bridging fluctuating hydrodynamics and molecular dynamics simulations of fluids”, J. Chemical Physics, 130 (2009), 134111 | DOI

[7] Pandey A., Klar A., Tiwari S., “Mecshfree method for fluctuating hydrodynamics”, Math. and Comp. in Simulation, 2012, 2157–2166 | DOI | MR | Zbl

[8] Fabritiis G. D. et al., “Fluctuating hydrodynamic modeling of fluids at the nanoscale”, Phys. Rev. E, 75 (2007), 026307 | DOI

[9] Landau L. D., Lifshits E. M., Teoreticheskaya fizika, Uchebnoe posobie. V 10 t., v. VI, Gidrodinamika, Nauka, M., 1986 | MR

[10] Hirsch C., Numerical computation of internal and external flows, v. 1, Fundaments of numerical discretezation, Antony Rowe Ltd., Eastbourne, 2001 | Zbl

[11] Goloviznin V. M., Karabasov S. A., Kobrinskii I. M., “Balansno-kharakteristicheskie skhemy s razdelennymi konservativnymi i potokovymi peremennymi”, Matem. modelirovanie, 15:9 (2003), 29–48 | MR | Zbl

[12] Godunov S. K., “Raznostnyi metod chislennogo resheniya razryvnykh reshenii uravnenii gidrodinamiki”, Matem. sb., 47:89 (1959), 271–306 | MR | Zbl

[13] Pirozzoli S., “On spectral properties of shocj-capturing schemes”, J. Comput. Phys., 219 (2006), 489–497 | DOI | Zbl

[14] Fauconnier D., Dick F., “On the spectral and conservation properties of nonlinear discretezation operators”, J. comput. phys., 230 (2011), 4488–4518 | DOI | MR | Zbl

[15] Ferrari P. A., Fontes L. R. G., “Shock fluctuations in the abymetric simple exchusion process”, Probab. Theory Relat. Fields, 99:205 (1994) | MR | Zbl