Extended Lagrangian approach for the numerical study of multidimensional dispersive waves: applications to the Serre-Green-Naghdi equations. (English) Zbl 07652801
Summary: In this paper we study two multidimensional nonlinear dispersive systems: the Serre-Green-Naghdi (SGN) equations describing dispersive shallow water flows, and the Iordanskii-Kogarko-Wijngaarden (IKW) equations describing fluids containing small compressible gas bubbles. These models are Euler-Lagrange equations for a given Lagrangian and share common mathematical structure, namely the dependence of the pressure on material derivatives of macroscopic variables. We develop a generic dispersive model such that SGN and IKW systems become its special cases if only one specifies the appropriate Lagrangian, and then use the extended Lagragian approach proposed in [N. Favrie and S. Gavrilyuk, Nonlinearity 30, No. 7, 2718–2736 (2017; Zbl 1432.65120)] to build its hyperbolic approximation. The new approximate model is unconditionally hyperbolic for both SGN and IKW cases, and accurately describes dispersive phenomena, which allows to impose discontinuous initial data and study dispersive shock waves. We consider the 2-D hyperbolic version of SGN system as an example for numerical simulations and apply a second order implicit-explicit scheme in order to numerically integrate the system. The obtained 1-D and 2-D results are in close agreement with available exact solutions and numerical tests.
MSC:
| 65Mxx | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 76Bxx | Incompressible inviscid fluids |
| 76Mxx | Basic methods in fluid mechanics |
Keywords:
dispersive shallow water equations; bubbly fluids; Euler-Lagrange equations; hyperbolic conservation laws; multidimensional waves; implicit-explicit numerical methodsCitations:
Zbl 1432.65120References:
| [1] | Ascher, U. M.; Ruuth, S. J.; Spiteri, R. J., Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Appl. Numer. Math., 25, 151-167 (1997) · Zbl 0896.65061 |
| [2] | Ascher, U. M.; Ruuth, S. J.; Wetton, B. T.R., Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal., 32, 3, 797-823 (1995) · Zbl 0841.65081 |
| [3] | Bedford, A.; Drumheller, D. S., A variational theory of immiscible mixtures, Arch. Ration. Mech. Anal., 68, 1, 37-51 (1978) · Zbl 0391.76012 |
| [4] | Berdichevsky, V. L., Variational Principles of Continuum Mechanics. I. Fundamentals (2009), Springer Science & Business Media · Zbl 1183.49002 |
| [5] | Bonneton, P.; Barthelemy, E.; Chazel, F.; Cienfuegos, R.; Lannes, D.; Marche, F.; Tissier, M., Recent advances in Serre-Green Naghdi modelling for wave transformation, breaking and runup processes, Eur. J. Mech. B, Fluids, 30, 6, 589-597 (2011) · Zbl 1258.76033 |
| [6] | Bonneton, P.; Chazel, F.; Lannes, D.; Marche, F.; Tissier, M., A splitting approach for the fully nonlinear and weakly dispersive Green-Naghdi model, J. Comput. Phys., 230, 4, 1479-1498 (2011) · Zbl 1391.76066 |
| [7] | Busto, S.; Dumbser, M.; Escalante, C.; Favrie, N.; Gavrilyuk, S., On high order ADER discontinuous Galerkin schemes for first order hyperbolic reformulations of nonlinear dispersive systems, J. Sci. Comput., 87, 48 (2021) · Zbl 1465.76060 |
| [8] | Chazel, F.; Lannes, D.; Marche, F., Numerical simulation of strongly nonlinear and dispersive waves using a Green-Naghdi model, J. Sci. Comput., 48, 105-116 (2011) · Zbl 1419.76454 |
| [9] | Colella, P., Multidimensional upwind methods for hyperbolic conservation laws, J. Comput. Phys., 87, 171-200 (1990) · Zbl 0694.65041 |
| [10] | Czarnota, C.; Molinari, A.; Mercier, S., The structure of steady shock waves in porous metals, J. Mech. Phys. Solids, 107, 204-228 (2017) |
| [11] | Dafermos, C. M., Hyperbolic Conservation Laws in Continuum Physics (2000), Springer: Springer Berlin · Zbl 0940.35002 |
| [12] | Davis, S. F., Simplified second-order Godunov-type methods, SIAM J. Sci. Stat. Comput., 9, 3, 445-473 (1988) · Zbl 0645.65050 |
| [13] | Dhaouadi, F., An augmented Lagrangian approach for Euler-Korteweg type equations (2020), Université Paul Sabatier - Toulouse 3, PhD thesis |
| [14] | Dhaouadi, F.; Favrie, N.; Gavrilyuk, S., Extended Lagrangian approach for the defocusing nonlinear Schrödinger equation, Stud. Appl. Math., 207, 1-23 (2018) |
| [15] | Duchêne, V., Rigorous justification of the Favrie-Gavrilyuk approximation to the Serre-Green-Naghdi model, Nonlinearity, 32, 3772-3797 (2019) · Zbl 1421.35208 |
| [16] | Duran, A.; Marche, F., Discontinuous-Galerkin discretization of a new class of Green-Naghdi equations, Commun. Comput. Phys., 17, 3, 721-760 (2015) · Zbl 1373.76082 |
| [17] | El, G. A.; Grimshaw, R. H.J.; Smyth, N. F., Unsteady undular bores in fully nonlinear shallow-water theory, Phys. Fluids, 18, Article 027104 pp. (2006) · Zbl 1185.76454 |
| [18] | El, G. A.; Hoefer, M. A., Dispersive shock waves and modulation theory, Phys. D, Nonlinear Phenom., 333, 11-65 (2016) · Zbl 1415.76001 |
| [19] | El, G. A.; Khodorovskii, V. V.; Tyurina, A. V., Undular bore transition in bi-directional conservative wave dynamics, Phys. D, Nonlinear Phenom., 206, 232-251 (2005) · Zbl 1073.35161 |
| [20] | Engel, P.; Viorel, A.; Rohde, C., A low-order approximation for viscous-capillary phase transition dynamics, Port. Math., 70, 4, 319-344 (2013) · Zbl 1302.35249 |
| [21] | Favrie, N.; Gavrilyuk, S., A rapid numerical method for solving Serre-Green-Naghdi equations describing long free surface gravity waves, Nonlinearity, 30, 7, 2718-2736 (2017) · Zbl 1432.65120 |
| [22] | Gavrilyuk, S., Multiphase flow modeling via Hamilton’s principle, (dell’Isola, F.; Gavrilyuk, S., Variational Models and Methods in Solid and Fluid Mechanics. Variational Models and Methods in Solid and Fluid Mechanics, CISM Courses and Lectures (2011), Springer Edition) · Zbl 1247.76085 |
| [23] | Gavrilyuk, S.; Nkonga, B.; Shyue, K.-M.; Truskinovsky, L., Stationary shock-like transition fronts in dispersive systems, Nonlinearity, 33, 10, 5477-5509 (2020) · Zbl 1452.35101 |
| [24] | Gavrilyuk, S.; Shugrin, S., Media with equations of state that depend on derivatives, J. Appl. Mech. Tech. Phys., 37, 2, 177-189 (1996) · Zbl 1031.76501 |
| [25] | Gavrilyuk, S. L.; Teshukov, V. M., Linear stability of parallel inviscid flows of shallow water and bubbly fluid, Stud. Appl. Math., 113, 1, 1-29 (2004) · Zbl 1141.76382 |
| [26] | Giesselmann, J.; Lattanzio, C.; Tzavaras, A. E., Relative energy for the Korteweg theory and related Hamiltonian flows in gas dynamics, Arch. Ration. Mech. Anal., 223, 3, 1427-1484 (2017) · Zbl 1359.35140 |
| [27] | Godunov, S. K.; Zabrodin, A. V.; Ivanov, M. Ya.; Kraiko, A. N.; Prokopov, G. P., Numerical Solution of Multidimensional Problems of Gas Dynamics (1976), Nauka Edition: Nauka Edition Moscow, (in Russian) |
| [28] | Green, A. E.; Laws, N.; Naghdi, P. M., On the theory of water waves, Proc. R. Soc. Lond., 338, 43-55 (1974) · Zbl 0289.76010 |
| [29] | Green, A. E.; Naghdi, P. M., A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech., 78, 2, 237-246 (1976) · Zbl 0351.76014 |
| [30] | Guermond, J. L.; Popov, B.; Tovar, E.; Kees, C., Robust explicit relaxation technique for solving the Green-Naghdi equations, J. Comput. Phys., 399, Article 108917 pp. (2019) · Zbl 1453.65321 |
| [31] | Iordanskii, S., On the equations of liquid motion with gas bubbles, Zh. Prikl. Meh. Teh. Fiz., 3, 102-110 (1960) |
| [32] | Kazakova, M.; Noble, P., Discrete transparent boundary conditions for the linearized Green-Naghdi system of equations, SIAM J. Numer. Anal., 58, 1, 657-683 (2020) · Zbl 1434.65121 |
| [33] | Ketcheson, D. I.; LeVeque, R. J., WENOCLAW: a higher order wave propagation method, (Benzoni-Gavage, S.; Serre, D., Hyperbolic Probl. Theory, Numer. Appl (2008), Springer: Springer Berlin, Heidelberg), 609-616 · Zbl 1138.65084 |
| [34] | Ketcheson, D. I.; Parsani, M.; LeVeque, R. J., High-order wave propagation algorithms for hyperbolic systems, SIAM J. Sci. Comput., 35, 1, A351-A377 (2013) · Zbl 1264.65151 |
| [35] | Kogarko, B. S., On a model of a cavitating liquid, Dokl. Akad. Nauk SSSR, 137, 6, 1331-1333 (1961) |
| [36] | Lannes, D.; Marche, F., A new class of fully nonlinear and weakly dispersive Green-Naghdi models for efficient 2D simulations, J. Comput. Phys., 282, 238-268 (2015) · Zbl 1351.76114 |
| [37] | Lax, P. D., Hyperbolic Partial Differential Equations, vol. 14 (2006), American Mathematical Soc. · Zbl 1113.35002 |
| [38] | Le Métayer, O.; Gavrilyuk, S.; Hank, S., A numerical scheme for the Green-Naghdi model, J. Comput. Phys., 229, 6, 2034-2045 (2010) · Zbl 1303.76105 |
| [39] | Li, M.; Guyenne, P.; Li, F.; Xu, L., High order well-balanced CDG-FE methods for shallow water waves by a Green-Naghdi model, J. Comput. Phys., 257, 169-192 (2014) · Zbl 1349.76233 |
| [40] | Liapidevskii, V. Yu.; Gavrilova, K. N., Dispersion and blockage effects in the flow over a sill, J. Appl. Mech. Tech. Phys., 49, 1, 34-45 (2008) · Zbl 1271.76045 |
| [41] | Mitsotakis, D.; Ilan, B.; Dutykh, D., On the Galerkin/finite-element method for the Serre equations, J. Sci. Comput., 61, 1, 166-195 (2014) · Zbl 1299.76026 |
| [42] | Nadiga, B. T.; Margolin, L. G.; Smolarkiewicz, P. K., Different approximations of shallow fluid flow over an obstacle, Phys. Fluids, 8, 8, 2066-2077 (1996) · Zbl 1082.76509 |
| [43] | Neusser, G.; Rohde, C.; Schleper, V., Relaxation of the Navier-Stokes-Korteweg equations for compressible two-phase flow with phase transition, Int. J. Numer. Methods Fluids, 79, 615-639 (2015) · Zbl 1455.65176 |
| [44] | Pareschi, L.; Russo, G., Implicit-explicit Runge-Kutta schemes for stiff systems of differential equations, Recent Trends Numer. Anal., 3, 269-288 (2001) · Zbl 1018.65093 |
| [45] | Pareschi, L.; Russo, G., Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation, Appl. Numer. Math., 25, 129-155 (2005) · Zbl 1203.65111 |
| [46] | Richard, G. L., An extension of the Boussinesq-type models to weakly compressible flows, Eur. J. Mech. B, Fluids, 89, 217-240 (2021) · Zbl 1501.76071 |
| [47] | Rusanov, V. V., The calculation of the interaction of non-stationary shock waves and obstacles, USSR Comput. Math. Math. Phys., 1, 2, 267-279 (1961) |
| [48] | Serre, F., Contribution à l’étude des écoulements permanents et variables dans les canaux, Houille Blanche, 3, 374-388 (1953) |
| [49] | Su, C. H.; Gardner, C. S., Korteweg-de Vries equation and generalizations. III. Derivation of the Korteweg-de Vries equation and Burgers equation, J. Math. Phys., 10, 3, 536-539 (1969) · Zbl 0283.35020 |
| [50] | Tkachenko, S., Analytical and numerical study of a dispersive shallow water model (2020), Aix-Marseille Université, PhD thesis |
| [51] | Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics: a Practical Introduction (2009), Springer Science & Business Media · Zbl 1227.76006 |
| [52] | Toro, E. F., The HLLC Riemann solver, Shock Waves, 29, 8, 1065-1082 (2019) |
| [53] | Toro, E. F.; Spruce, M.; Speares, W., Restoration of the contact surface in the HLL-Riemann solver, Shock Waves, 4, 1, 25-34 (1994) · Zbl 0811.76053 |
| [54] | Whitham, G. B., Linear and Nonlinear Waves, A Wiley-Interscience Series of Texts, Monographs, and Tracts (1974) · Zbl 0373.76001 |
| [55] | Wijngaarden, L. V., On the equations of motion for mixtures of liquid and gas bubbles, J. Fluid Mech., 33, 3, 465-474 (1968) · Zbl 0187.52202 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
