Volume 21, Issue 4
Schemes with Well-Controlled Dissipation. Hyperbolic Systems in Nonconservative Form

Abdelaziz Beljadid, Philippe G. LeFloch, Siddhartha Mishra & Carlos Pares

Commun. Comput. Phys., 21 (2017), pp. 913-946.

Published online: 2018-04

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  • 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 smallscale 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.

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@Article{CiCP-21-913, author = {Abdelaziz Beljadid, Philippe G. LeFloch, Siddhartha Mishra and Carlos Pares}, title = {Schemes with Well-Controlled Dissipation. Hyperbolic Systems in Nonconservative Form}, journal = {Communications in Computational Physics}, year = {2018}, volume = {21}, number = {4}, pages = {913--946}, 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 smallscale 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2016-0019}, url = {http://global-sci.org/intro/article_detail/cicp/11266.html} }
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