A new class of entropy stable schemes for hyperbolic systems: Finite element methods
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- by Ioannis Gkanis and Charalambos G. Makridakis;
- Math. Comp. 90 (2021), 1663-1699
- DOI: https://doi.org/10.1090/mcom/3617
- Published electronically: March 17, 2021
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Abstract:
In this work we propose a new class of entropy consistent schemes for hyperbolic systems of conservation laws (HCL). The schemes developed so far in the classic works of Tadmor, Johnson and their collaborators start from an appropriate entropy conservative formulation of the system. Then entropy diminishing schemes are obtained by adding appropriate artificial diffusion terms. This program was based on the formulation of the HCL using the entropy variables. In this work we propose an alternative approach which has as a starting point a new mixed reformulation of the hyperbolic system which retains the original variables but still allows for conservative discretisation. The original variables are approximated directly and significant flexibility is allowed in the design of the corresponding computational algorithms. New finite element schemes are introduced and analysed. It is shown that the resulting approximations are consistent at the limit to an entropy weak and when appropriate to an entropy measure valued solution.References
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Bibliographic Information
- Ioannis Gkanis
- Affiliation: Department of Mathematics, MPS, University of Sussex, Brighton BN1 9QH, United Kingdom
- Email: i.gkanis@sussex.ac.uk
- Charalambos G. Makridakis
- Affiliation: Modelling and Scientific Computing, DMAM, University of Crete/Institute of Applied and Computational Mathematics-FORTH, GR 70013 Heraklion, Greece; and MPS, University of Sussex, Brighton BN1 9QH, United Kingdom
- MR Author ID: 289627
- Email: C.G.Makridakis@iacm.forth.gr
- Received by editor(s): September 18, 2019
- Received by editor(s) in revised form: July 31, 2020, October 28, 2020, and November 9, 2020
- Published electronically: March 17, 2021
- Additional Notes: This work was partially supported by the EU Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie project ModCompShock (modcompshock.eu) agreement No. 642768.
- © Copyright 2021 American Mathematical Society
- Journal: Math. Comp. 90 (2021), 1663-1699
- MSC (2020): Primary 65N12
- DOI: https://doi.org/10.1090/mcom/3617
- MathSciNet review: 4273112