• G. V. Sandrakov Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
Keywords: discretization, conservation laws, particle-in-cell, large particles, phase transitions


Computational algorithms for modeling of multiphase hydrodynamics processes with take of phase transitions will be discussed. The algorithms are based on discretization of conservation laws for mass, momentum, and energy in integral and differential forms. The time and spatial discretization is natural and numerical simulations are realized as direct computer experiments. The experiments are implemented as a computer simulation of the dynamics of a multiphase carrier fluid containing particles that can undergo, for example, graphite–diamond phase transitions and calculations are given. Modification of the algorithms have also been developed to take into account the influence of viscosity when simulating the dynamics of a multiphase fluid in porous media.


Nigmatulin R. I. Dynamics of multiphase media. New York: Hemisphere, 1991.

Yudovich V. I. Eleven great problems of mathematical hydrodynamics. Mosc. Math. J. 2003. Vol. 3. P. 711–737.

Sandrakov G. V., Boyko S. B. Mathematical modeling of complex heterogeneous fluid dynamics. Journal of Numerical and Applied Mathematics. 2011. № 1 (104). P. 109–120.

Boyko S. B., Sandrakov G. V. Mathematical modeling of phase transitions graphite-diamond dynamics. Journal of Numerical and Applied Mathematics. 2012. № 2 (108). P. 88–109.

Sandrakov G. V. Multiphase models of nonstationary diffusion arising from homogenization. Comp. Math and Math Physics. 2004. Vol. 44(10). P. 1741–1756.

Sandrakov G. V. Multiphase homogenized diffusion models for problems with several parameters. Izvestiya Mathematics. 2007. Vol. 71(6). P. 1193–1252.

Sandrakov G. Modeling of heterogeneous hydrodynamics processes with phase transition. Modeling, Control and Information Technologies. 2019. Vol. 3. P. 67–68.

Sandrakov G. A modified method for modeling of heterogeneous hydrodynamics processes. Modeling, Control and Information Technologies. 2020. Vol. 4. P. 63–66.

Harlow F. Numerical particles-in-cells method. In: Fundamental methods in hydridinamics. New York: Academic Press, 1964.

Belotserkovskij O. M., Davidov Yu. M. The large particle method in gas dynamics. Moscow: Nauka, 1982.

Grigoriev Yu. N., Vshivkov V. A., Fedoruk M. P. Numerical Particle-In-Cell Methods: Theory and Applications. Boston: Utrecht, 2002.

Bakhvalov N. S., Panasenko G. Homogenisation: averaging processes in periodic media. Springer Netherlands: Kluwer Academic Publishers, 1989.

Meyer K. Physics-mechanical crystallography. Moscow: Metallurgy, 1972. (In Russian)

Baum F. A., Orlenko L. P., Stanyukovich K. P., Chelyshev V. P., Shekhter B. I. Physics of an explosion. Moscow: Nauka, 1975.

Danilenko V. V. Synthesis and sintering of diamond by explosion. Moscow: Energoatomizdat, 2003.

Danilenko V. V. From the history of the discovery of nanodiamond synthesis. Solid state physics. 2006. Vol. 46:4. P. 581–584. (In Russian)

Boyko S. B., Mischenko V. V., Sandrakov G. V. The numerical investigation method for evaporated plasma. Journal of Numerical and Applied Mathematics. 2007. № 2 (95). P. 3–12.

Dikalyuk A. S., Kuratov S. E. Numerical modeling of plasma devices by the particle-in-cell method on unstructured grids. Math. Models and Computer Simulations. 2018. Vol. 10. P. 198–208.

Kormann K., Sonnendrucker E. Energy-conserving time propagation for a structure-preserving particle-in-cell Vlasov-Maxwell solver. J. Comput. Physics. 2021. Vol. 425. 109890.

Wang Z., Qin H., Sturdevant B., Chang C. S. Geometric electrostatic particlein-cell algorithm on unstructured meshes. J. Plasma Physics. 2021. Vol. 87(4). 905870406.

Parreiras E. A., Vieira M. B., Machado A. G., Renhe M. C., Giraldi G. A. A particle-in-cell method for anisotropic fluid simulation. Computers and Graphics. 2022. Vol. 102. P. 220–232.

Liu G. R., Liu M. B.: Smoothed particle hydrodynamics: a meshfree particle method. New Jersey: World Scientific Publishing, 2003.

Liu M. B., Liu G. R.: Smoothed particle hydrodynamics (SPH): an overview and recent developments. Arch. Comput. Method Eng. 2010. Vol. 17. P. 25–76.

Wang Z. B., Chen R., Wang H., Liao Q., Zhu X., Li S. Z. An overview of smoothed particle hydrodynamics for simulating multiphase flow. Applied Math. Modelling. 2016. Vol. 40(23-24). P. 9625–9655.

Koukouvinis P., Kyriazis N., Gavaises M. Smoothed particle hydrodynamics simulation of a laser pulse impact onto a liquid metal droplet. PLoS ONE. 2018. Vol. 13(9). e0204125.

Britun V. F., Kurdyumov A. V., Borimchuk N. I. , Yarosh V. V. Estimation of the P,T-conditions realized under high-temperature shock compression of boron nitride in a cylindrical storage ampoule. Powder metallurgy. 2007. No. 1. P. 3–10. (In Russian)

Lomonosov I. V., Fortov V. E., Frolova A. A., Khishchenko K. V., Charakhchyan A. A., Shurshalov L. V. Numerical study of shock compression of graphite and its transformation into diamond in conical targets. Journal of Technical Physics. 2003. Vol. 73:6. P. 66–75. (In Russian)

Belkheeva R. K. Simulation of direct and reverse phase transitions under shock-wave loading of graphite. Siberian Journal of Industrial Mathematics. 2007. Vol. 10:1. P. 25–32. (In Russian)

Li S., Liu W. K.: A meshfree particle method. Berlin: Springer-Verlag, 2004.

Lions P.-L. Mathematical topics in fluid mechanics. Vol. 2. Oxford: Clarendon Press, 1998.