Reduced basis techniques for nonlinear conservation laws
Taddei, T.1 ; Perotto, S.2 ; Quarteroni, A.2, 3
1 Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Ave, Rm. 3-237 Cambridge, MA 02139, USA.
2 MOX-Modeling and Scientific Computing, Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy.
3 Modelling and Scientific Computing, Institute of Analysis and Scientific Computing, École Polytechnique Féderale de Lausanne, Station 8, EPFL, 1015 Lausanne, Switzerland.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 3, pp. 787-814.

In this paper we present a new reduced basis technique for parametrized nonlinear scalar conservation laws in presence of shocks. The essential ingredients are an efficient algorithm to approximate the shock curve, a procedure to detect the smooth components of the solution at the two sides of the shock, and a suitable interpolation strategy to reconstruct such smooth components during the online stage. The approach we propose is based on some theoretical properties of the solution to the problem. Some numerical examples prove the effectiveness of the proposed strategy.

Reçu le :
DOI : 10.1051/m2an/2014054
Classification : 65M08, 65M15, 65M25, 35L65, 35L67
Mots-clés : Nonlinear conservation laws, model reduction, reduced basis method
Taddei, T. 1 ; Perotto, S. 2 ; Quarteroni, A. 2, 3

1 Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Ave, Rm. 3-237 Cambridge, MA 02139, USA.
2 MOX-Modeling and Scientific Computing, Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy.
3 Modelling and Scientific Computing, Institute of Analysis and Scientific Computing, École Polytechnique Féderale de Lausanne, Station 8, EPFL, 1015 Lausanne, Switzerland. 
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Taddei, T.; Perotto, S.; Quarteroni, A. Reduced basis techniques for nonlinear conservation laws. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 3, pp. 787-814. doi : 10.1051/m2an/2014054. http://archive.numdam.org/articles/10.1051/m2an/2014054/

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