A combined finite volume - finite element scheme for a low-Mach system involving a Joule term

Caterina Calgaro; Claire Colin; Emmanuel Creusé; Caterina Calgaro; Claire Colin; Emmanuel Creusé

doi:10.3934/math.2020021

AIMS Mathematics

2020, Volume 5, Issue 1: 311-331. doi: 10.3934/math.2020021

Previous Article Next Article

Research article Topical Sections

A combined finite volume - finite element scheme for a low-Mach system involving a Joule term

1.
University Lille, CNRS, UMR 8524, Inria - Laboratoire Paul Painlevé, F-59000 Lille, France
2.
University Polytechnique Hauts-de-France, EA 4015, LAMAV - FR CNRS 2956, F-59313 Valenciennes, France

Received: 16 July 2019 Accepted: 07 November 2019 Published: 13 November 2019
MSC : 35Q30, 35Q35, 65M08, 65M60

In this paper, we propose a combined finite volume - finite element scheme, for the resolution of a specific low-Mach model expressed in the velocity, pressure and temperature variables. The dynamic viscosity of the fluid is given by an explicit function of the temperature, leading to the presence of a so-called Joule term in the mass conservation equation. First, we prove a discrete maximum principle for the temperature. Second, the numerical fluxes defined for the finite volume computation of the temperature are efficiently derived from the discrete finite element velocity field obtained by the resolution of the momentum equation. Several numerical tests are presented to illustrate our theoretical results and to underline the efficiency of the scheme in term of convergence rates.
- low Mach model,
- finite volume scheme,
- finite element scheme,
- Joule term,
- maximum principle
Citation: Caterina Calgaro, Claire Colin, Emmanuel Creusé. A combined finite volume - finite element scheme for a low-Mach system involving a Joule term[J]. AIMS Mathematics, 2020, 5(1): 311-331. doi: 10.3934/math.2020021

Related Papers:

Abstract

In this paper, we propose a combined finite volume - finite element scheme, for the resolution of a specific low-Mach model expressed in the velocity, pressure and temperature variables. The dynamic viscosity of the fluid is given by an explicit function of the temperature, leading to the presence of a so-called Joule term in the mass conservation equation. First, we prove a discrete maximum principle for the temperature. Second, the numerical fluxes defined for the finite volume computation of the temperature are efficiently derived from the discrete finite element velocity field obtained by the resolution of the momentum equation. Several numerical tests are presented to illustrate our theoretical results and to underline the efficiency of the scheme in term of convergence rates.

References

[1]	M. Avila, J. Principe and R. Codina, A finite element dynamical nonlinear subscale approximation for the low Mach number flow equations, J. Comput. Phys., 230 (2011), 7988-8009. doi: 10.1016/j.jcp.2011.06.032
[2]	A. Beccantini, E. Studer, S. Gounand, et al. Numerical simulations of a transient injection flow at low Mach number regime, Int. J. Numer. Meth. Eng., 76 (2008), 662-696. doi: 10.1002/nme.2331
[3]	A. Bradji and R. Herbin, Discretization of coupled heat and electrical diffusion problems by finiteelement and finite-volume methods, IMA J. Numer. Anal., 28 (2008), 469-495.
[4]	D. Bresch, E. H. Essoufi and M. Sy, Effect of density dependent viscosities on multiphasic incompressible fluid models, J. Math. Fluid Mech., 9 (2007), 377-397. doi: 10.1007/s00021-005-0204-4
[5]	D. Bresch, V. Giovangigli and E. Zatorska, Two-velocity hydrodynamics in fluid mechanics: Part I Well posedness for zero Mach number systems, J. Math. Pure. Appl., 104 (2015), 762-800. doi: 10.1016/j.matpur.2015.05.003
[6]	C. Calgaro, E. Chane-Kane, E. Creusé, et al. L^∞-stability of vertex-based MUSCL finite volume schemes on unstructured grids: simulation of incompressible flows with high density ratios, J. Comput. Phys., 229 (2010), 6027-6046. doi: 10.1016/j.jcp.2010.04.034
[7]	C. Calgaro, C. Colin and E. Creusé, A combined finite volumes - finite elements method for a low-Mach model, Int. J. Numer. Meth. Fl., 90 (2019), 1-21. doi: 10.1002/fld.4706
[8]	C. Calgaro, C. Colin, E. Creusé, et al. Approximation by an iterative method of a low-Mach model with temperature dependant viscosity, Math. Method. Appl. Sci., 42 (2019), 250-271. doi: 10.1002/mma.5342
[9]	C. Calgaro, E. Creusé and T. Goudon, An hybrid finite volume-finite element method for variable density incompressible flows, J. Comput. Phys., 227 (2008), 4671-4696. doi: 10.1016/j.jcp.2008.01.017
[10]	C. Calgaro, E. Creusé and T. Goudon, Modeling and simulation of mixture flows: application to powder-snow avalanches, Comput. Fluids, 107 (2015), 100-122. doi: 10.1016/j.compfluid.2014.10.008
[11]	C. Calgaro, M. Ezzoug and E. Zahrouni, Stability and convergence of an hybrid finite volumefinite element method for a multiphasic incompressible fluid model, Commun. Pure Appl. Anal., 17 (2018), 429-448. doi: 10.3934/cpaa.2018024
[12]	C. Cancès and C. Guichard, Convergence of a nonlinear entropy diminishing control volume finite element scheme for solving anisotropic degenerate parabolic equations, Math. Comput., 85 (2016), 549-580.
[13]	C. Chainais-Hillairet, Discrete duality finite volume schemes for two-dimensional drift-diffusion and energy-transport models, Int. J. Numer. Meth. Fl., 59 (2009), 239-257. doi: 10.1002/fld.1393
[14]	C. Chainais-Hillairet, Y.-J. Peng and I. Violet, Numerical solutions of Euler-Poisson systems for potential flows, Appl. Numer. Math., 59 (2009), 301-315. doi: 10.1016/j.apnum.2008.02.006
[15]	D. R. Chenoweth and S. Paolucci, Natural convection in an enclosed vertical air layer with large horizontal temperature differences, J. Fluid Mech., 169 (1986), 173-210. doi: 10.1017/S0022112086000587
[16]	P. G. Ciarlet, Introduction to numerical linear algebra and optimisation, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 1989.
[17]	C. Colin, Analyse et simulation numérique par méthode combinée Volumes Finis - Eléments Finis de modèles de type Faible Mach, PhD thesis, Université de Lille, 2019.
[18]	Y. Coudière, J.-P. Vila and P. Villedieu, Convergence rate of a finite volume scheme for a two-dimensional convection-diffusion problem, ESAIM: Mathematical Modelling and Numerical Analysis, 33 (1999), 493-516. doi: 10.1051/m2an:1999149
[19]	R. Danchin and X. Liao, On the well-posedness of the full low Mach number limit system in general critical Besov spaces, Commun. Contemp. Math., 14 (2012), 1250022.
[20]	S. Dellacherie, On a diphasic low Mach number system, ESAIM: Mathematical Modelling and Numerical Analysis, 39 (2005), 487-514. doi: 10.1051/m2an:2005020
[21]	K. Domelevo and P. Omnes, A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids, ESAIM: Mathematical Modelling and Numerical Analysis, 39 (2005), 1203-1249. doi: 10.1051/m2an:2005047
[22]	J. Droniou and R. Eymard, A mixed finite volume scheme for anisotropic diffusion problems on any grid, Numer. Math., 105 (2006), 35-71. doi: 10.1007/s00211-006-0034-1
[23]	P. Embid, Well-posedness of the nonlinear equations for zero Mach number combustion, Commun. Part. Diff. Eq., 12 (1987), 1227-1283. doi: 10.1080/03605308708820526
[24]	R. Eymard and T. Gallouët, H-convergence and numerical schemes for elliptic problems, SIAM J. Numer. Anal., 41 (2003), 539-562. doi: 10.1137/S0036142901397083
[25]	R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, Handbook of numerical analysis, 7 (2000), 713-1018.
[26]	R. Eymard, T. Gallouët and R. Herbin, A cell-centered finite-volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension, IMA J. Numer. Anal., 26 (2006), 326-353. doi: 10.1093/imanum/dri036
[27]	R. Herbin, J.-C. Latché and K. Saleh, Low Mach number limit of a pressure correction MAC scheme for compressible barotropic flows. In: Finite volumes for complex applications VIII- methods and theoretical aspects, volume 199 of Springer Proc. Math. Stat., pages 255-263. Springer, Cham, 2017.
[28]	R. Herbin, J.-C. Latché and K. Saleh, Low Mach number limit of some staggered schemes for compressible barotropic flows, arXiv preprint, arXiv:1803.09568.
[29]	V. Heuveline, On higher-order mixed FEM for low Mach number flows: application to a natural convection benchmark problem, Int. J. Numer. Meth. Fl., 41 (2003), 1339-1356. doi: 10.1002/fld.454
[30]	F. Huang and W. Tan, On the strong solution of the ghost effect system, SIAM J. Math. Anal., 49 (2017), 3496-3526. doi: 10.1137/16M106964X
[31]	P. Le Quéré, C. Weisman, H. Paillère, et al. Modelling of natural convection flows with large temperature differences: a benchmark problem for low Mach number solvers. Part I. Reference solutions, ESAIM: Mathematical Modelling and Numerical Analysis, 39 (2005), 609-616. doi: 10.1051/m2an:2005027
[32]	C. D. Levermore, W. Sun and K. Trivisa, Local well-posedness of a ghost effect system, Indiana Univ. Math. J., 60 (2011), 517-576. doi: 10.1512/iumj.2011.60.4179
[33]	P.-L. Lions, Mathematical topics in fluid mechanics: Volume 2: Compressible models, volume 10 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York, 1998.
[34]	A. Majda and J. Sethian, The derivation and numerical solution of the equations for zero Mach number combustion, Combust. Sci. Technol., 42 (1985), 185-205. doi: 10.1080/00102208508960376

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)