Skip to main content Accessibility help
Hostname: page-component-55b6f6c457-qgndx Total loading time: 0.243 Render date: 2021-09-27T20:47:25.787Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

Numerical Approximation of a Compressible Multiphase System

Published online by Cambridge University Press:  03 June 2015

Remi Abgrall*
INRIA and Institut de Mathématiques de Bordeaux, Institut Polytechnique de Bordeaux, 200 route de la Vieille Tour, 33 405 Talence, France
Harish Kumar*
Department of Mathematics, IIT Delhi, New Delhi, India-110016
Get access


The numerical simulation of non conservative system is a difficult challenge for two reasons at least. The first one is that it is not possible to derive jump relations directly from conservation principles, so that in general, if the model description is non ambiguous for smooth solutions, this is no longer the case for discontinuous solutions. From the numerical view point, this leads to the following situation: if a scheme is stable, its limit for mesh convergence will depend on its dissipative structure. This is well known since at least [1]. In this paper we are interested in the “dual” problem: given a system in non conservative form and consistent jump relations, how can we construct a numerical scheme that will, for mesh convergence, provide limit solutions that are the exact solution of the problem. In order to investigate this problem, we consider a multiphase flow model for which jump relations are known. Our scheme is an hybridation of Glimm scheme and Roe scheme.

Research Article
Copyright © Global Science Press Limited 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


[1]Karni, S.Viscous shock profiles and primitive formulations. SIAM Journal on Numerical Analysis, 29(1592160), 1992.CrossRefGoogle Scholar
[2]Kapila, A.K., Menikoff, R., Bdzil, J.B., Son, S.F., and Stewart, D.S.Two-phase modeling of deflagration-to-detonation transition in granular materials: reduced equations. Phys. Fluids, 13(10):23, 2001.CrossRefGoogle Scholar
[3]Baer, M.R. and Nunziato, J.W.A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials. Int. J. Multiphase Flow, 12:861889, 1986.CrossRefGoogle Scholar
[4]Murrone, A. and Guillard, H.A five equation reduced model for compressible two-phase flow problems. J. Comput. Phys., 202(2):664698,2005.CrossRefGoogle Scholar
[5]Saurel, R., Le Métayer, O., Massoni, J., and Gavrilyuk, S.Shock jump relations for multiphase mixtures with stiff mechanical relaxation. Shock Waves, 16(3):209232, 2007.CrossRefGoogle Scholar
[6]Abgrall, R. and Karni, S.A comment on the computation of non-conservative products. J. Comput. Phys., 229(8):27592763, 2010.CrossRefGoogle Scholar
[7]Godlewski, E. and Raviart, P.-A.Numerical approximation of hyperbolic systems of conservation laws. New York, NY: Springer, 1996.CrossRefGoogle Scholar
[8]Petipas, F., Franquet, E., Saurel, R., and Le Métayer, O.A relaxation–projection method for compressible flows, part II: artificial heat exchange for multiphase shocks. J. Comput Phys., 223(2):822845, 2007.Google Scholar
[9]Petitpas, F., Saurel, R., Franquet, E, and Chinnayya, A.Modelling detonation waves in condensed energetic materials: multiphase cj conditions and multidimensional computations. Shock Waves, 19(5):377401, 2009.CrossRefGoogle Scholar
[10] Manuel Castro, J.; LeFloch, Philippe G.; Luz Muñoz-Ruiz, María and Parés, Carlos. Why many theories of shock waves are necessary: convergence error in formally path-consistent schemes. J. Comput. Phys., 227(17):81078129,2008.CrossRefGoogle Scholar
[11]Gallice, G.Positive and entropy stable Godunov schemes for gas dynamics and MHD equations in Lagrangian or Elerian coordinates. Numer. Math., 94:673713, 2003.CrossRefGoogle Scholar
[12]Gallice, G.Roe’s matrices for general conservation laws in Eulerian or Lagrangian coordinates. application to gas dynamics and MHD. C.R. Acad. Sci. Paris, 321:10691072, 1995.Google Scholar
[13]Roe, P.L.Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys., 43:357372, 1981.CrossRefGoogle Scholar
[14]Roe, P.L. Fluctuations and signals - A framework for numerical evolution problems, 1982.Google Scholar
[15]Abgrall, R.Residual distribution schemes: current status and future trends. Comput. Fluids, 35(7):641669, 2006.CrossRefGoogle Scholar
[16]Pares, C.Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J. Numer. Anal., 44:300321, 2006.CrossRefGoogle Scholar
[17]Sweby, P.K.High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal., 21(5), 1984.CrossRefGoogle Scholar
[18]Karni, S.Multi-component flow calculations by a consistent primitive algorithm. Journal of Computational Physics, 112:3143, 1994.CrossRefGoogle Scholar
[19]Fjordholm, U.S. and Mishra, S.Accurate numerical discretizations of non-conservative hyperbolic systems. Technical Report 2010-25, ETH Zürich, Seminar für Applied mathematisch, 2010.Google Scholar
[20]Castro, M.J., Fjordholm, U.S., Mishra, S., and Parés., C.Entropy stable conservative and entropy stable scheme for non-conservative hyperbolic systems. Technical Report 2011-49, Seminar für Angewandte Mathematik, ETH Zürich, 2011.Google Scholar
[21]Collela, P.Glimm’s method for gas dynamic. SIAM J. Sci. Stat. Comput., 3(1):76110, 1982.CrossRefGoogle Scholar
[22]Sod, G.A.A numerical study of a converging cylindrical shock. J. Fluid Mech., 83(785794), 1977.CrossRefGoogle Scholar
[23]Glimm, J., Marshall, G., and Plohr, B.A generalized Riemann problem for quasi-one dimensional gas flows. Advances in Applied Mathematics, 5:130, 1984.CrossRefGoogle Scholar
[24]Le Martelot, S., Nkonga, B., and Saurel, R.Liquid and liquid-gas flows at all speeds. Journal of Computational Physics, 255:5382, 2013.CrossRefGoogle Scholar
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Numerical Approximation of a Compressible Multiphase System
Available formats

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Numerical Approximation of a Compressible Multiphase System
Available formats

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Numerical Approximation of a Compressible Multiphase System
Available formats

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *