A vector Hamilton–Jacobi formulation for the numerical simulation of Euler flows

Philippe Hoch; Olivier Pironneau

doi:10.1016/j.crma.2005.11.007

Numerical Analysis

A vector Hamilton–Jacobi formulation for the numerical simulation of Euler flows

Presented by: Olivier Pironneau

Philippe Hoch ¹; Olivier Pironneau ²

¹ CEA/DAM Île de France, service DCSA/SSEL, BP 12, 91680 Bruyères le Châtel, France
² Laboratoire Jacques-Louis Lions, UPMC, 175, rue du Chevaleret, 75013 Paris, France

Comptes Rendus. Mathématique, Volume 342 (2006) no. 2, pp. 151-156.

Abstract
Résumé

A vector Hamilton–Jacobi formulation of the Euler equations for fluids is studied numerically. The long term objective is to find the sensitivity of a flow with respect to a parameter, which is solution of the linearized Euler equations with Dirac singularities in the initial conditions. A Hamilton–Jacobi formulation uses integral of the primitive variable so that Dirac singularities become shocks. It is shown here that there are vector Hamilton–Jacobi formulations for any vector conservation laws and that they can be simulated numerically with packages such as GO++ which we adapted to the vector case both on structured and unstructured meshes for this purpose.

On présente, à des fins numériques, une formulation de type Hamilton–Jacobi vectoriel pour les lois de conservations comme les équations d'Euler pour les fluides compressibles. L'application visée est le calcul des sensibilités des écoulements par rapport à un paramètre car il faut alors résoudre une équation d'Euler linéarisée avec des masses de Dirac dans les conditions initiales alors qu'avec la formulation Hamilton–Jacobi les masses de Dirac deviennent des discontinuités de chocs. On montre ici que toute loi de conservation vectorielle admet une représentation Hamilton–Jacobi vectorielle et que ces nouvelles équations peuvent être intégrées numériquement par les techniques du logiciel GO++ par exemple, qui a été adapté aux cas vectoriels en maillage structuré et non-structuré pour cet objectif.

Received: 2005-05-16
Accepted: 2005-11-07
Published online: 2005-12-15

DOI: 10.1016/j.crma.2005.11.007

Author's affiliations:

Philippe Hoch ¹; Olivier Pironneau ²

¹ CEA/DAM Île de France, service DCSA/SSEL, BP 12, 91680 Bruyères le Châtel, France
² Laboratoire Jacques-Louis Lions, UPMC, 175, rue du Chevaleret, 75013 Paris, France

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     author = {Philippe Hoch and Olivier Pironneau},
     title = {A vector {Hamilton{\textendash}Jacobi} formulation for the numerical simulation of {Euler} flows},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {151--156},
     publisher = {Elsevier},
     volume = {342},
     number = {2},
     year = {2006},
     doi = {10.1016/j.crma.2005.11.007},
     language = {en},
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Philippe Hoch; Olivier Pironneau. A vector Hamilton–Jacobi formulation for the numerical simulation of Euler flows. Comptes Rendus. Mathématique, Volume 342 (2006) no. 2, pp. 151-156. doi : 10.1016/j.crma.2005.11.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.11.007/

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