Schemes with well-controlled dissipation. Hyperbolic systems in nonconservative form

Abdelaziz Beljadid (1, 2) , Philippe G Lefloch (2) , Siddhartha Mishra (3) , Carlos Parés (4)

1 MIT - Massachusetts Institute of Technology
2 LJLL - Laboratoire Jacques-Louis Lions
3 ETH-SAM - Seminar for Applied Mathematics
4 Universidad de Málaga [Málaga] = University of Málaga [Málaga]

We propose here a class of numerical schemes for the approximation of weak solutions to nonlinear hyper-bolic systems in nonconservative form —the notion of solution being understood in the sense of Dal Maso, LeFloch, and Murat (DLM). The proposed numerical method falls within LeFloch-Mishra's framework of schemes with well-controlled dissipation (WCD), recently introduced for dealing with small-scale dependent shocks. We design WCD schemes which are consistent with a given nonconservative system at arbitrarily high-order and then analyze their linear stability. We then investigate several nonconservative hyperbolic models arising in complex fluid dynamics, and we numerically demonstrate the convergence of our schemes toward physically meaningful weak solutions.

Domaines: Equations aux dérivées partielles [math.AP]

Abdelaziz Beljadid, Philippe G Lefloch, Siddhartha Mishra, Carlos Parés. Schemes with well-controlled dissipation. Hyperbolic systems in nonconservative form. 2016. ⟨hal-01423827⟩