Commun. Comput. Phys., 15 (2014), pp. 1237-1265.


Numerical Approximation of a Compressible Multiphase System

Remi Abgrall 1*, Harish Kumar 2

1 INRIA and Institut de Mathematiques de Bordeaux, Institut Polytechnique de Bordeaux, 200 route de la Vieille Tour, 33 405 Talence, France.
2 Department of Mathematics, IIT Delhi, New Delhi, India-110016.

Received 11 March 2013; Accepted (in revised version) 23 September 2013
Available online 30 January 2014
doi:10.4208/cicp.110313.230913a

Abstract

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.

AMS subject classifications: 65M06, 65M08, 65M12, 35L60, 35L65, 35L67

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Key words: Non conservative systems, numerical approximation, Glimm's scheme, Roe's scheme.

*Corresponding author.
Email: remi.abgrall@inria.fr (R. Abgrall), hkumar@maths.iitd.ac.in (H. Kumar)
 

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