Лагранжев метод для жестких задач динамики двухфазной среды с релаксацией: частица-сетка или частица-частица
DOI:
https://doi.org/10.26089/NumMet.v26r215Ключевые слова:
равномерные численные методы, двухжидкостная гидродинамика сглаженных частиц, SPH-IDIC, динамика газовзвесей, жесткие релаксационные слагаемые, жесткое трениеАннотация
Макроуровневые модели динамики газовзвесей часто представляют собой дифференциальные уравнения в частных производных с релаксационными слагаемыми, описывающими передачу импульса и энергии от газа к частицам и наоборот. Для ультрадисперсных частиц время релаксации намного короче, чем время, на котором рассматривается динамика среды. В работе исследуется лагранжев метод моделирования динамики газовзвесей “двухжидкостная гидродинамика сглаженных частиц” (Two-Fluid Smoothed Particle Hydrodynamics, TFSPH). TFSPH подразумевает, что каждая фаза (газ и частицы) моделируется своим набором частиц. В рамках TFSPH известны два подхода к расчету релаксационного взаимодействия (трения), которое определяется разностью скоростей между несущей и дисперсной фазами: частица-частица и частица-сетка. Ранее в вычислительных экспериментах было установлено, что для малых времен релаксации в подходе частица-частица имеет место избыточная диссипация волн, а подход частица-сетка свободен от этого недостатка. В работе впервые дано объяснение этому явлению средствами вычислительной математики.
Библиографические ссылки
V. P. Maslov, Asymptotic Methods and Perturbation Theory(Nauka, Moscow, 1988; Springer, New York, 1999).
I. V. Andrianov and L. I. Manevitch, Asymptotology: Ideas, Methods, and Applications(Aslan, Moscow, 1994; Kluwer, Dordrecht, 2002).
M. I. Vishik and L. A. Lyusternik, “Regular Degeneration and Boundary Layer for Linear Differential Equations with Small Parameter,” Usp. Mat. Nauk 12 (5), 3-122 (1957) [Russ. Math. Surv. 12 (5), 3-122 (1957)].
A. L. Goldenveizer, Theory of Elastic Thin Shells(Nauka, Moscow, 1976; Pergamon Press, New York, 1961).
A. B. Vasil’eva and V. F. Butuzov, Asymptotic Expansions of Solutions of Singularly Perturbed Equations(Nauka, Moscow, 1973) [in Russian].
A. B. Vasil’eva and V. F. Butuzov, Asymptotic Methods in the Theory of Singular Perturbations(Vysshaya Shkola, Moscow, 1990) [in Russian].
A. Nayfeh, Perturbation Methods(Wiley, New York, 1973; Mir, Moscow, 1976).
S. A. Lomov, Introduction to the General Theory of Singular Perturbations(Nauka, Moscow, 1981; Am. Math. Soc. Vol. 112, Providence, 1992).
E. P. Doolan, J. J. H. Miller, and W. H. A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers(Boole Press, Dublin, 1980; Mir, Moscow, 1983).
L. Govindarao, Parameter Uniform Numerical Methods for Singularly Perturbed Parabolic Partial Differential Equations , PhD Thesis in Mathematics and Statistics (National Institute of Technology, Rourkela, India, 2019).
N. N. Nefedov and N. N. Deryugina, “The Existence of a Boundary-Layer Stationary Solution to a Reaction-Diffusion Equation with Singularly Perturbed Neumann Boundary Condition,” Vestn. Mosk. Univ., Ser. 3: Fiz. Astron., No. 5, 30-34 (2020) [Moscow Univ. Phys. Bull. 75 (5), 409-414 (2020)].
doi 10.3103/S0027134920050185
A. V. Nesterov and O. V. Shuliko, “ Asymptotic Behavior of the Solution of a Weakly Nonlinear System of Differential Equations of “Reaction-Diffusion” Type,” Mat. Model. 16 (8), 50-58 (2004).
A. I. Zadorin, Finite-Difference Schemes for Nonlinear Differential Equations with a Small Parameter in Bounded and Unbounded Domains , Doctoral Thesis in Physics and Mathematics (Inst. Comput. Math. Math. Geophys., Novosibirsk, 2000).
V. D. Liseikin, Grid Generation Methods (Springer, Cham, 2017).
doi 10.1007/978-3-319-57846-0
N. V. Kopteva, A Posteriori and a Priori Estimates of Finite Element Solutions of Some Singularly Perturbed Equations on Anisotropic Meshes. Doctoral Thesis in Physics and Mathematics (University of Limerik, Ireland, 2018).
https://www.keldysh.ru/council/3/D00202403/kopteva_diss.pdf . Cited May 17, 2025.
M. V. Alekseev, “Numerical Algorithms for Solving Two-Phase Flows Based on Relaxation Baer-Nunziato Model,” Numerical Methods and Programming. 24 (2), 182-194 (2023).
doi 10.26089/NumMet.v24r214
V. N. Snytnikov, E. E. Peskova, and O. P. Stoyanovskaya, “Mathematical Model of a Two-Temperature Medium of Gas-Solid Nanoparticles with Laser Methane Pyrolysis,” Mat. Model. 35 (4), 24-50 (2023) [Math. Models Comput. Simul. 15 (5), 877-893 (2023)].
doi 10.1134/S2070048223050095
V. Akimkin, E. Vorobyov, Y. Pavlyuchenkov, and O. Stoyanovskaya, “Gravitoviscous Protoplanetary Discs with a Dust Component -- IV. Disc Outer Edges, Spectral Indices, and Opacity Gaps,” Mon. Not. R. Astron. Soc. 499 (4), 5578-5597 (2020).
doi 10.1093/mnras/staa3134
E. I. Vorobyov, V. G. Elbakyan, A. Johansen, et al., “Formation of Pebbles in (Gravito-)Viscous Protoplanetary Disks with Various Turbulent Strengths,” Astron. Astrophys. 670. Article Number A81 (2023).
doi 10.1051/0004-6361/202244500
S. S. Khrapov, “Nonlinear Dynamics of Acoustic Instability in a Vibrationally Excited Gas: Influence of Relaxation Time and Structure of Shock Waves,” Physical-Chemical Kinetics in Gas Dynamics. 25 (7) (2024).
http://chemphys.edu.ru/issues/2024-25-7/articles/1151 . Cited May 17, 2025.
A. G. Petrova, “Asymptotic Analysis of Viscoelastic Fluid Models with Two Small Relaxation Parameters,” Zh. Prikl. Mekh. Tekh. Fiz. 65, No. 5, 157-168 (2024) [J. Appl. Mech. Tech. Phys. 65 (5), 933-943 (2024)].
doi 10.1134/S0021894424050146
T.-P. Liu, “Hyperbolic Conservation Laws with Relaxation,” Commun. Math. Phys. 108, 153-175 (1987).
doi 10.1007/BF01210707
F. E. Marble, “Dynamics of Dusty Gases,” Annu. Rev. Fluid Mech. 2, 397-446 (1970).
doi 10.1146/annurev.fl.02.010170.002145
D. V. Sadin, “A Method for Computing Heterogeneous Wave Flows with Intense Phase Interaction,” Zh. Vychisl. Mat. Mat. Fiz. 38, N 6, 1033-1039 (1998) [Comput. Math. Math. Phys. 38 (6), 987-993 (1998)].
S. Jin, “Asymptotic Preserving (AP) Schemes for Multiscale Kinetic and Hyperbolic Equations: A Review,” in Lecture Notes for Summer School on Methods and Models of Kinetic Theory (M&MKT) , Porto Ercole (Grosseto, Italy). 2010. 177-216.
P. Degond and F. Deluzet, “Asymptotic-Preserving Methods and Multiscale Models for Plasma Physics,” J. Comput. Phys. 336, 429-457 (2017).
doi 10.1016/j.jcp.2017.02.009
G. Laibe and D. J. Price, “Dusty Gas with Smoothed Particle Hydrodynamics -- I. Algorithm and Test Suite,” Mon. Not. R. Astron. Soc. 420 (3), 2345-2364 (2012).
doi 10.1111/j.1365-2966.2011.20202.x
J. J. Monaghan, “Smoothed Particle Hydrodynamics,” Rep. Prog. Phys. 68 (8), 1703-1759 (2005).
doi 10.1088/0034-4885/68/8/R01
T. Demidova, T. Savvateeva, S. Anoshin, et al., “Implementation of Dusty Gas Model Based on Fast and Implicit Particle-Mesh Approach SPH-IDIC in Open-Source Astrophysical Code GADGET-2,” in Lecture Notes in Computer Science (Springer, Cham, 2023), Vol. 14389, pp. 195-208.
doi 10.1007/978-3-031-49435-2_14
O. P. Stoyanovskaya, N. V. Snytnikov, and V. N. Snytnikov, “An Algorithm for Solving Transient Problems of Gravitational Gas Dynamics: A Combination of the SPH Method with a Grid Method of Gravitational Potential Computation,” Numerical Methods and Programming. 16 (1), 52-60 (2015).
doi 10.26089/NumMet.v16r106
O. P. Stoyanovskaya, “Numerical Simulation of Gravitational Instability Development and Clump Formation in Massive Circumstellar Disks Using Integral Characteristics for the Interpretation of Results,” Numerical Methods and Programming. 17 (3), 339-352 (2016).
doi 10.26089/NumMet.v17r332
V. Springel, “The Cosmological Simulation Code GADGET-2,” Mon. Not. R. Astron. Soc. 364 (4), 1105-1134 (2005).
doi 10.1111/j.1365-2966.2005.09655.x
D. J. Price, J. Wurster, T. S. Tricco, et al., “Phantom: A Smoothed Particle Hydrodynamics and Magnetohydrodynamics Code for Astrophysics,” Publ. Astron. Soc. Aust. 35, id.e031 (2018).
doi 10.1017/pasa.2018.25
C. Zhang, M. Rezavand, Y. Zhu, et al., “SPHinXsys: An Open-Source Multi-Physics and Multi-Resolution Library Based on Smoothed Particle Hydrodynamics,” Comput. Phys. Commun. 267, Article Number 108066 (2021).
doi 10.1016/j.cpc.2021.108066
J. Monaghan and A. Kocharyan, “SPH Simulation of Multi-Phase Flow,” Comput. Phys. Commun. 87 (1-2), 225-235 (1995).
doi 10.1016/0010-4655(94)00174-Z
O. P. Stoyanovskaya, T. A. Glushko, N. V. Snytnikov, and V. N. Snytnikov, “Two-Fluid Dusty Gas in Smoothed Particle Hydrodynamics: Fast and Implicit Algorithm for Stiff Linear Drag,” Astron. Comput. 25, 25-37 (2018).
doi 10.1016/j.ascom.2018.08.004
O. Stoyanovskaya, M. Davydov, M. Arendarenko, et al., “Fast Method to Simulate Dynamics of Two-Phase Medium with Intense Interaction between Phases by Smoothed Particle Hydrodynamics: Gas-Dust Mixture with Polydisperse Particles, Linear Drag, One-Dimensional Tests,” J. Comput. Phys. 430, Article Number 110035 (2021).
doi 10.1016/j.jcp.2020.110035
J. P. Morris, “A Study of the Stability Properties of Smooth Particle Hydrodynamics,” Publ. Astron. Soc. Aust. 13 (1), 97-102 (1996).
doi 10.1017/S1323358000020610
S.-H. Cha and A. P. Whitworth, “Implementations and Tests of Godunov-Type Particle Hydrodynamics,” Mon. Not. R. Astron. Soc. 340 (1), 73-90 (2003).
doi 10.1046/j.1365-8711.2003.06266.x
W. Dehnen and H. Aly, “Improving Convergence in Smoothed Particle Hydrodynamics Simulations without Pairing Instability,” Mon. Not. R. Astron. Soc. 425 (2), 1068-1082 (2012).
doi 10.1111/j.1365-2966.2012.21439.x
O. P. Stoyanovskaya, V. V. Lisitsa, S. A. Anoshin, et al., “Dispersion Analysis of SPH as a Way to Understand Its Order of Approximation,” J. Comput. Appl. Math. 438, Article Number 115495 (2024).
doi 10.1016/j.cam.2023.115495
T. V. Markelova, M. S. Arendarenko, E. A. Isaenko, and O. P. Stoyanovskaya, “Plane Sound Waves of Small Amplitude in a Gas-Dust Mixture with Polydisperse Particles,” Zh. Prikl. Mekh. Tekh. Fiz. 62, No. 4. 158-168 (2021) [J. Appl. Mech. Tech. Phys. 62 (4), 663-672 (2021)].
doi 10.1134/S0021894421040167
O. P. Stoyanovskaya, V. V. Grigoryev, T. A. Savvateeva, et al., “Multi-Fluid Dynamical Model of Isothermal Gas and Buoyant Dispersed Particles: Monodisperse Mixture, Reference Solution of DustyWave Problem as Test for CFD-Solvers, Effective Sound Speed for High and Low Mutual Drag,” Int. J. Multiph. Flow 149, Article Number 103935 (2022).
doi 10.1016/j.ijmultiphaseflow.2021.103935
T. V. Markelova and O. P. Stoyanovskaya, “Plane Sound Waves in a Macroscopic Model of a Two-Velocity Two-Temperature Gas-Dust Mixture,” J. Appl. Mech. Tech. Phys. 2025 (in press).
O. P. Stoyanovskaya, G. D. Turova, and N. M. Yudina, “Dispersion and Group Analysis of Dusty Burgers Equations,” Lobachevskii J. Math. 45 (1), 108-118 (2024).
doi 10.1134/s1995080224010505
O. P. Stoyanovskaya, O. A. Burmistrova, M. S. Arendarenko, and T. V. Markelova, “Dispersion Analysis of SPH for Parabolic Equations: High-Order Kernels against Tensile Instability,” J. Comput. Appl. Math. 457, Article Number 116316 (2025).
doi 10.1016/j.cam.2024.116316
R. Fatehi and M. T. Manzari, “Error Estimation in Smoothed Particle Hydrodynamics and a New Scheme for Second Derivatives,” Comput. Math. Appl. 61 (2), 482-498 (2011).
doi 10.1016/j.camwa.2010.11.028
G. Laibe and D. J. Price, “Dusty Gas with Smoothed Particle Hydrodynamics -- II. Implicit Timestepping and Astrophysical Drag Regimes,” Mon. Not. R. Astron. Soc. 420 (3), 2365-2376 (2012).
doi 10.1111/j.1365-2966.2011.20201.x
P. Lorén-Aguilar and M. R. Bate, “Two-Fluid Dust and Gas Mixtures in Smoothed Particle Hydrodynamics II: an Improved Semi-Implicit Approach,” Mon. Not. R. Astron. Soc. 454 (4), 4114-4119 (2015).
doi 10.1093/mnras/stv2262
R. A. Booth, D. Sijacki, and C. J. Clarke, “Smoothed Particle Hydrodynamics Simulations of Gas and Dust Mixtures,” Mon. Not. R. Astron. Soc. 452 (4), 3932-3947 (2015).
doi 10.1093/mnras/stv1486
M. Hutchison, D. J. Price, and G. Laibe, “MULTIGRAIN: a Smoothed Particle Hydrodynamic Algorithm for Multiple Small Dust Grains and Gas,” Mon. Not. R. Astron. Soc. 476 (2), 2186-2198 (2018).
doi 10.1093/mnras/sty367
D. J. Price and G. Laibe, “A Solution to the Overdamping Problem when Simulating Dust-Gas Mixtures with Smoothed Particle Hydrodynamics,” Mon. Not. R. Astron. Soc. 495 (4), 3929-3934 (2020).
doi 10.1093/mnras/staa1366
D. Elsender and M. R. Bate, “An Implicit Algorithm for Simulating the Dynamics of Small Dust Grains with Smoothed Particle Hydrodynamics,” Mon. Not. R. Astron. Soc. 529 (4), 4455-4467 (2024).
doi 10.1093/mnras/stae722
O. Burmistrova, T. Markelova, M. Arendarenko, and O. Stoyanovskaya, “A New Method for Approximating of First Derivatives in Smoothed Particle Hydrodynamics: Theory and Practice for Linear Transport Equation,” Lobachevskii J. Math. 46, 43-54 (2025).
doi 10.1134/S1995080224608312
M. Arendarenko, A. Dzhanbekova, S. Kotov, et al., “SymDR: Symbol Computer Algebra Library for Generation of Classical and Approximate Dispersion Relations for Systems of Partial Differential Equations,” Lobachevskii J. Math. 46, 1-12 (2025).
doi 10.1134/S1995080224608579
Загрузки
Опубликован
Выпуск
Раздел
Лицензия
Copyright (c) 2025 О. П. Стояновская
Это произведение доступно по лицензии Creative Commons «Attribution» («Атрибуция») 4.0 Всемирная.
