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

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

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 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.

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@Article{CiCP-21-913, author = {Abdelaziz Beljadid , and Philippe G. LeFloch , and Siddhartha Mishra , and Carlos Parés , }, 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 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2016-0019}, url = {http://global-sci.org/intro/article_detail/cicp/11266.html} }
TY - JOUR T1 - Schemes with Well-Controlled Dissipation. Hyperbolic Systems in Nonconservative Form AU - Abdelaziz Beljadid , AU - Philippe G. LeFloch , AU - Siddhartha Mishra , AU - Carlos Parés , JO - Communications in Computational Physics VL - 4 SP - 913 EP - 946 PY - 2018 DA - 2018/04 SN - 21 DO - http://doi.org/10.4208/cicp.OA-2016-0019 UR - https://global-sci.org/intro/article_detail/cicp/11266.html KW - AB -

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.

Abdelaziz Beljadid, Philippe G. LeFloch, Siddhartha Mishra & Carlos Parés. (2020). Schemes with Well-Controlled Dissipation. Hyperbolic Systems in Nonconservative Form. Communications in Computational Physics. 21 (4). 913-946. doi:10.4208/cicp.OA-2016-0019
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