Schemes with Well-Controlled Dissipation. Hyperbolic Systems in Nonconservative Form. (8th March 2017)

• Record Type: Journal Article
• Authors: Beljadid, Abdelaziz; LeFloch, Philippe G.; Mishra, Siddhartha; Parés, Carlos
• Is Part Of: Communications in computational physics. Volume 21:Number 4(2017:Apr.)
• Page Start: 913
• Page End: 946
• Publication Date: 2017-03-08
• Subjects: 35L65, -- 65M06, -- 76N15; Nonlinear hyperbolic, -- nonconservative system, -- discrete scheme, -- controlled dissipation, -- complex fluid dynamics; Mathematical physics -- Data processing -- Periodicals; Physics -- Data processing -- Periodicals; 530.150285
• DOI: 10.4208/cicp.OA-2016-0019
• Languages: English
• ISSNs: 1815-2406

Abstract

Abstract: We propose here a class of numerical schemes for the approximation of weak solutions to nonlinear hyperbolic 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.