Abstract

This paper is devoted to a centered IMEX scheme in a multidimensional framework for a wide class of multicomponent and isentropic flows. The proposed strategy is based on a regularized model where the advection velocity is modified by the gradient of the potential of the conservative forces in both mass and momentum equations. The stability of the scheme is ensured by the dissipation of mechanic energy, which stands for a mathematical entropy, under an advective CFL condition. The main physical properties, such as positivity, conservation of the total momentum, and conservation of the steady state at rest, are satisfied. In addition, asymptotic preserving properties in the regimes (``incompressible" and “acoustic") are analyzed. Finally, several simulations are presented to illustrate our results in a simplified context of oceanic flows in one dimension.

Keywords

  1. conservation laws
  2. low-Mach number
  3. low-Froude number
  4. well-balanced scheme
  5. entropy dissipation
  6. asymptotic preserving

MSC codes

  1. 35L60
  2. 76M12
  3. 86A05
  4. 76E20
  5. 35B40

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References

1.
E. Audusse, A multilayer Saint-Venant model: Derivation and numerical validation, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), pp. 189--214, https://doi.org/10.3934/dcdsb.2005.5.189.
2.
E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein, and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. Comput., 25 (2004), pp. 2050--2065, https://doi.org/10.1137/S1064827503431090.
3.
A. Bedford and D. S. Drumheller, Theories of immiscible and structured mixtures, Internat. J. Engrg. Sci., 21 (1983), pp. 863--960, https://doi.org/10.1016/0020-7225(83)90071-X.
4.
A. Bermudez and M. E. Vazquez, Upwind methods for hyperbolic conservation laws with source terms, Comput. & Fluids, 23 (1994), pp. 1049--1071, https://doi.org/10.1016/0045-7930(94)90004-3.
5.
M. Bernard, S. Dellacherie, G. Faccanoni, B. Grec, and Y. Penel, Study of a low mach nuclear core model for two-phase flows with phase transition I: Stiffened gas law, ESAIM Math. Model. Numer. Anal., 48 (2014), pp. 1639--1679, https://doi.org/10.1051/m2an/2014015.
6.
S. Bianchini, On the Riemann problem for non-conservative hyperbolic systems, Arch. Ration. Mech. Anal., 166 (2003), pp. 1--26, http://link.springer.com/article/10.1007/s00205-002-0227-4.
7.
F. Bouchut and T. M. de Luna, An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment, ESAIM Math. Model. Numer. Anal., 42 (2008), pp. 683--698, https://doi.org/10.1051/m2an:2008019.
8.
M. Castro, J. M. Gallardo, J. A. López-García, and C. Parés, Well-balanced high order extensions of Godunov's method for semilinear balance laws, SIAM J. Numer. Anal., 46 (2008), pp. 1012--1039, https://doi.org/10.1137/060674879.
9.
M. J. Castro, E. D. Fernández-Nieto, A. M. Ferreiro, J. A. García-Rodríguez, and C. Parés, High order extensions of Roe schemes for two-dimensional nonconservative hyperbolic systems, J. Sci. Comput., 39 (2009), pp. 67--114, https://doi.org/10.1007/s10915-008-9250-4.
10.
C. Chainais-Hillairet, Y.-J. Peng, and I. Violet, Numerical solutions of Euler--Poisson systems for potential flows, Appl. Numer. Math., 59 (2009), pp. 301--315, https://doi.org/10.1016/j.apnum.2008.02.006.
11.
A.-J.-C. B. de Saint-Venant, Théorie du mouvement non permanent des eaux, avec application aux crues des rivières et à l'introduction des marées dans leurs lits, C.R. Acad. Sci. Paris, 73 (1871), pp. 147--154.
12.
S. Dellacherie, Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number, J. Comput. Phys., 229 (2010), pp. 978--1016, https://doi.org/10.1016/j.jcp.2009.09.044.
13.
V. Duchêne, Asymptotic shallow water models for internal waves in a two-fluid system with a free surface, SIAM J. Math. Anal., 42 (2010), pp. 2229--2260, https://doi.org/10.1137/090761100.
14.
W. Gangbo and M. Westdickenberg, Optimal transport for the system of isentropic Euler equations, Comm. Partial Differential Equations, 34 (2009), pp. 1041--1073, https://doi.org/10.1080/03605300902892345.
15.
S. Gavrilyuk and H. Gouin, A new form of governing equations of fluids arising from Hamilton's principle, Internat. J. Engrg. Sci., 37 (1999), pp. 1495--1520, https://doi.org/10.1016/S0020-7225(98)00131-1.
16.
E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Appl. Math. Sci. 118, Springer-Verlag, New York, 1996.
17.
J. M. Greenberg and A. Y. Leroux, A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal., 33 (1996), pp. 1--16, https://doi.org/10.1137/0733001.
18.
N. Grenier, J.-P. Vila, and P. Villedieu, An accurate low-Mach scheme for a compressible two-fluid model applied to free-surface flows, J. Comput. Phys., 252 (2013), pp. 1--19, https://doi.org/10.1016/j.jcp.2013.06.008.
19.
H. Guillard and C. Viozat, On the behaviour of upwind schemes in the low Mach number limit, Comput. & Fluids, 28 (1999), pp. 63--86, https://doi.org/10.1016/S0045-7930(98)00017-6.
20.
M. Ilcak, A. J. Adcroft, S. M. Griffies, and R. W. Hallberg, Spurious dianeutral mixing and the role of momentum closure, Ocean Model., 45/46 (2012), pp. 37--58, https://doi.org/10.1016/j.ocemod.2011.10.003.
21.
S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., 21 (1999), pp. 441--454, https://doi.org/10.1137/S1064827598334599.
22.
S. Jin and X. Wen, Two interface-type numerical methods for computing hyperbolic systems with geometrical source terms having concentrations, SIAM J. Sci. Comput., 26 (2005), pp. 2079--2101, https://doi.org/10.1137/040605825.
23.
S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), pp. 481--524, https://doi.org/10.1002/cpa.3160340405.
24.
S. Klainerman and A. Majda, Compressible and incompressible fluids, Comm. Pure Appl. Math., 35 (1982), pp. 629--651, https://doi.org/10.1002/cpa.3160350503.
25.
P. Lax and B. Wendroff, Systems of conservation laws, Comm. Pure Appl. Math., 13 (1960), pp. 217--237, https://doi.org/10.1002/cpa.3160130205.
26.
M.-S. Liou and C. J. Steffen, Jr., A new flux splitting scheme, J. Comput. Phys., 107 (1993), pp. 23--39, https://doi.org/10.1006/jcph.1993.1122.
27.
R. Liska and B. Wendroff, Analysis and computation with stratified fluid models, J. Comput. Phys., 137 (1997), pp. 212--244, https://doi.org/10.1006/jcph.1997.5806.
28.
R. R. Long, Long waves in a two-fluid system, J. Meteorology, 13 (1956), pp. 70--74, https://doi.org/10.1175/1520-0469(1956)013% 3C0070:LWIATF% 3E2.0.CO;2.
29.
G. Métivier and S. Schochet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal., 158 (2001), pp. 61--90, https://doi.org/10.1007/PL00004241.
30.
R. Monjarret, Local Well-Posedness of the Multi-layer Shallow Water Model with Free Surface, preprint, https://arxiv.org/abs/1411.2342v2, 2014.
31.
R. Monjarret, Local well-posedness of the two-layer shallow water model with free surface, SIAM J. Appl. Math., 75 (2015), pp. 2311--2332, https://doi.org/10.1137/140957020.
32.
M. L. Mun͂oz-Ruiz and C. Parés, Some comments on the numerical approximation of hyperbolic nonconservative systems, in Numerical Methods for Hyperbolic Equations: Theory and Applications, CRC Press/Balkema, Leiden, The Netherlands, 2013, pp. 311--318, https://doi.org/10.1201/b14172-43.
33.
R. Natalini, M. Ribot, and M. Twarogowska, A well-balanced numerical scheme for a one dimensional quasilinear hyperbolic model of chemotaxis, Commun. Math. Sci., 12 (2014), pp. 13--39, https://doi.org/10.4310/CMS.2014.v12.n1.a2.
34.
S. Noelle, N. Pankratz, G. Puppo, and J. R. Natvig, Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows, J. Comput. Phys., 213 (2006), pp. 474--499, https://doi.org/10.1016/j.jcp.2005.08.019.
35.
S. Noelle, Y. Xing, and C.-W. Shu, High-order well-balanced finite volume WENO schemes for shallow water equation with moving water, J. Comput. Phys., 226 (2007), pp. 29--58, https://doi.org/10.1016/j.jcp.2007.03.031.
36.
L. Ovsyannikov, Two-layer “shallow water” model, J. Appl. Mech. Tech. Phys., 20 (1979), pp. 127--135, https://doi.org/10.1007/BF00910010.
37.
M. Parisot and J.-P. Vila, Numerical scheme for multilayer shallow-water model in the low-Froude number regime, C. R. Math. Acad. Sci. Paris, 352 (2014), pp. 953--957, https://doi.org/10.1016/j.crma.2014.09.020.
38.
B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term, Calcolo, 38 (2001), pp. 201--231, https://doi.org/10.1007/s10092-001-8181-3.
39.
R. Rannacher, On Chorin's projection method for the incompressible Navier-Stokes equations, in the Navier-Stokes Equations II---Theory and Numerical Methods (Oberwolfach, 1991), Lecture Notes in Math. 1530, Springer, Berlin, 1992, pp. 167--183, https://doi.org/10.1007/BFb0090341.
40.
G. Reznik, V. Zeitlin, and M. Ben Jelloul, Nonlinear theory of geostrophic adjustment. Part 1. Rotating shallow-water model, J. Fluid Mech., 445 (2001), pp. 93--120, https://doi.org/10.1017/S002211200100550X.
41.
J. Serrin, Mathematical principles of classical fluid mechanics, in Handbuch der Physik (herausgegeben von S. Flügge), Bd. 8/1, Strömungsmechanik I (Mitherausgeber C. Truesdell), Springer-Verlag, Berlin, Göttingen, Heidelberg, 1959, pp. 125--263, https://doi.org/10.1007/978-3-642-45914-6_2.
42.
M. Slemrod, Dynamic phase transitions in a Van Der Waals fluid, J. Differential Equations, 52 (1984), pp. 1--23, https://doi.org/10.1016/0022-0396(84)90130-X.
43.
W. A. Strauss, Partial Differential Equations: An Introduction, John Wiley, New York, 1992.
44.
L. V. Wijngaarden, On the equations of motion for mixtures of liquid and gas bubbles, J. Fluid Mech., 33 (1968), pp. 465--474, https://doi.org/10.1017/S002211206800145X.

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Information

Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 3083 - 3104
ISSN (online): 1095-7170

History

Submitted: 19 May 2015
Accepted: 21 July 2016
Published online: 11 October 2016

Keywords

  1. conservation laws
  2. low-Mach number
  3. low-Froude number
  4. well-balanced scheme
  5. entropy dissipation
  6. asymptotic preserving

MSC codes

  1. 35L60
  2. 76M12
  3. 86A05
  4. 76E20
  5. 35B40

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