# Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes [Well-posedness of a singular balance law]

Abstract : We define entropy weak solutions and establish well-posedness for the Cauchy problem for the formal equation $\partial_t u(t,x) + \partial_x \frac{u^2}2(t,x) = - \lambda u(t,x) \delta_0(x),$ which can be seen as two Burgers equations coupled in a non-conservative way through the interface located at $x=0$. This problem appears as an important auxiliary step in the theoretical and numerical study of the one-dimensional particle-in-fluid model developed by Lagoutière, Seguin and Takahashi [LST08]. The interpretation of the non-conservative product $u(t,x) \delta_0(x)$'' follows the analysis of [LST08]; we can describe the associated interface coupling in terms of one-sided traces on the interface. Well-posedness is established using the tools of the theory of conservation laws with discontinuous flux ([AKR11]). For proving existence and for practical computation of solutions, we construct a finite volume scheme, which turns out to be a well-balanced scheme and which allows a simple and efficient treatment of the interface coupling. Numerical illustrations are given.
Keywords :
Type de document :
Article dans une revue
Domaine :

Littérature citée [30 références]

https://hal.archives-ouvertes.fr/hal-00576959
Contributeur : Boris Andreianov <>
Soumis le : mercredi 16 mars 2011 - 07:57:17
Dernière modification le : vendredi 4 janvier 2019 - 17:32:30
Document(s) archivé(s) le : jeudi 8 novembre 2012 - 11:56:20

### Fichier

AndrSeguin-preprint-2011.pdf
Fichiers produits par l'(les) auteur(s)

### Citation

Boris Andreianov, Nicolas Seguin. Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes [Well-posedness of a singular balance law]. DCDS-A, 2012, 32 (6), pp. 1939-1964. 〈https://www.aimsciences.org/journals/displayPaperReference.jsp?paperID=6983〉. 〈10.3934/dcds.2012.32.1939〉. 〈hal-00576959〉

### Métriques

Consultations de la notice

## 735

Téléchargements de fichiers