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A well-balanced discontinuous Galerkin method for the first--order Z4 formulation of the Einstein--Euler system

hal.structure.identifierUniversità degli Studi di Trento = University of Trento [UNITN]
dc.contributor.authorDUMBSER, Michael
hal.structure.identifierUniversità degli Studi di Trento = University of Trento [UNITN]
dc.contributor.authorZANOTTI, Olindo
hal.structure.identifierCertified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts [CARDAMOM]
dc.contributor.authorGABURRO, Elena
hal.structure.identifierUniversità degli Studi di Trento = University of Trento [UNITN]
dc.contributor.authorPESHKOV, Ilya
dc.date.accessioned2024-04-04T02:32:02Z
dc.date.available2024-04-04T02:32:02Z
dc.date.issued2023-07-13
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/190350
dc.description.abstractEnIn this paper we develop a new well-balanced discontinuous Galerkin (DG) finite element scheme with subcell finite volume (FV) limiter for the numerical solution of the Einstein--Euler equations of general relativity based on a first order hyperbolic reformulation of the Z4 formalism. The first order Z4 system, which is composed of 59 equations, is analyzed and proven to be strongly hyperbolic for a general metric. The well-balancing is achieved for arbitrary but a priori known equilibria by subtracting a discrete version of the equilibrium solution from the discretized time-dependent PDE system. Special care has also been taken in the design of the numerical viscosity so that the well-balancing property is achieved. As for the treatment of low density matter, e.g. when simulating massive compact objects like neutron stars surrounded by vacuum, we have introduced a new filter in the conversion from the conserved to the primitive variables, preventing superluminal velocities when the density drops below a certain threshold, and being potentially also very useful for the numerical investigation of highly rarefied relativistic astrophysical flows. Thanks to these improvements, all standard tests of numerical relativity are successfully reproduced, reaching three achievements: (i) we are able to obtain stable long term simulations of stationary black holes, including Kerr black holes with extreme spin, which after an initial perturbation return perfectly back to the equilibrium solution up to machine precision; (ii) a (standard) TOV star under perturbation is evolved in pure vacuum ($\rho=p=0$) up to $t=1000$ with no need to introduce any artificial atmosphere around the star; and, (iii) we solve the head on collision of two punctures black holes, that was previously considered un--tractable within the Z4 formalism.
dc.language.isoen
dc.subject.enEinstein field equations
dc.subject.enrelativistic Euler equations
dc.subject.enfirst order hyperbolic formulation of the Z4 formalism
dc.subject.endiscontinuous Galerkin
dc.subject.ennon conservative
dc.subject.enwell-balancing
dc.title.enA well-balanced discontinuous Galerkin method for the first--order Z4 formulation of the Einstein--Euler system
dc.typeDocument de travail - Pré-publication
dc.subject.halInformatique [cs]/Analyse numérique [cs.NA]
dc.identifier.arxiv2307.06629
dc.description.sponsorshipEuropeStructure Preserving schemes for Conservation Laws on Space Time Manifolds
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
hal.identifierhal-04341812
hal.version1
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-04341812v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.date=2023-07-13&rft.au=DUMBSER,%20Michael&ZANOTTI,%20Olindo&GABURRO,%20Elena&PESHKOV,%20Ilya&rft.genre=preprint


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