Hyperbolic Conservation Laws on Spacetimes

LeFloch, Philippe G.

doi:10.1007/978-1-4419-9554-4_21

Cited by 8 publications

(7 citation statements)

References 25 publications

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“…Hyperbolic conservation laws on curved spacetimes have been analyzed in recent years by LeFloch and collaborators; cf. the review [5], as well as [1,2,7]. In order to formulate the initial value problem for (1.1), we assume that the spacetime is foliated by hypersurfaces, that is, (1.2) in which each slice H t is an n-dimensional manifold, canonically endowed with a normal 1-form field N t and with the same topology as the one of the initial slice H 0 .…”

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confidence: 99%

Relativistic Burgers Equations on Curved Spacetimes. Derivation and Finite Volume Approximation

LeFloch¹,

Makhlof²,

Okutmustur³

2012

SIAM J. Numer. Anal.

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Within the class of nonlinear hyperbolic balance laws posed on a curved spacetime (endowed with a volume form), we identify a hyperbolic balance law that enjoys the same Lorentz invariance property as the one satisfied by the Euler equations of relativistic compressible fluids. This model is unique up to normalization and converges to the standard inviscid Burgers equation in the limit of infinite light speed. Furthermore, from the Euler system of relativistic compressible flows on a curved background, we derive both the standard inviscid Burgers equation and our relativistic generalizations. The proposed models are referred to as relativistic Burgers equations on curved spacetimes and provide us with simple models on which numerical methods can be developed and analyzed. Next, we introduce a finite volume scheme for the approximation of discontinuous solutions to these relativistic Burgers equations. Our scheme is formulated geometrically and is consistent with the natural divergence form of the balance laws under consideration. It applies to weak solutions containing shock waves and, most importantly, is well balanced in the sense that it preserves steady solutions. Numerical experiments are presented which demonstrate the convergence of the proposed finite volume scheme and its relevance for computing entropy solutions on a curved background.

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confidence: 99%

Relativistic Burgers Equations on Curved Spacetimes. Derivation and Finite Volume Approximation

LeFloch¹,

Makhlof²,

Okutmustur³

2012

SIAM J. Numer. Anal.

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“…Several studies have been recently developed for hyperbolic conservation laws posed on curved manifolds. The solutions of conservation laws including the systems on manifolds and on spacetimes were studied in [26,24] and by LeFloch and co-authors [1,2,6,5] and [10]- [11]. More recently, hyperbolic conservation laws for evolving surface were investigated by Dziuk, Kroöner and Müller [7], Giesselman [9], and Dziuk and Elliott [8].…”

Section: Introductionmentioning

confidence: 99%

A central-upwind geometry-preserving method for hyperbolic conservation laws on the sphere

Beljadid

LeFloch

2017

Commun. Appl. Math. Comput. Sci.

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We introduce a second-order, central-upwind finite volume method for the discretization of nonlinear hyperbolic conservation laws posed on the two-dimensional sphere. The semi-discrete version of the proposed method is based on a technique of local propagation speeds and it is free of any Riemann solver. The main advantages of our scheme are the high resolution of discontinuous solutions, its low numerical dissipation, and its simplicity for the implementation. The proposed scheme does not use any splitting approach, which is applied in some cases to upwind schemes in order to simplify the resolution of Riemann problems. The semi-discrete form of the scheme is strongly linked to the analytical properties of the nonlinear conservation law and to the geometry of the sphere. The curved geometry is treated here in an analytical way so that the semi-discrete form of the proposed scheme is consistent with a geometric compatibility property. Furthermore, the time evolution is carried out by using a total-variation-diminishing Runge-Kutta method. A rich family of (discontinuous) stationary solutions is available for the problem under consideration when the flux is nonlinear and foliated (as identified by the author in an earlier work). We present here a series of numerical examples, obtained by considering non-trivial steady state solutions and this leads us to a good validation of the accuracy and efficiency of the proposed central-upwind finite volume method. Our numerical tests confirm the stability of the proposed scheme and clearly show its ability to capture accurately discontinuous steady state solutions to nonlinear hyperbolic conservation laws posed on the sphere.

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“…In the first part, we defined this class of spacetimes and established an existence theory by posing the initial value problem from arbitrary initial data with weak regularity. The study of weakly regular spacetimes with symmetry was initiated in Christodoulou [3] (for vacuum spacetimes with radial symmetry) and LeFloch et al [7,9,10,11,15,16] (for vacuum or matter spacetimes with Gowdy symmetry). See also Rendall and Ståhl [18] for a proof that singularities arise in regular spacetimes.…”

Section: Introductionmentioning

confidence: 99%

Weakly regular T2-symmetric spacetimes. The future causal geometry of Gowdy spacetimes

LeFloch

Smulevici

2016

Journal of Differential Equations

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We investigate the future asymptotic behavior of Gowdy spacetimes on T 3 , when the metric satisfies weak regularity conditions, so that the metric coefficients (in suitable coordinates) are only in the Sobolev space H 1 or have even weaker regularity. The authors recently introduced this class of spacetimes in the broader context of T 2 -symmetric spacetimes and established the existence of a global foliation by spacelike hypersurfaces when the time function is chosen to be the area of the surfaces of symmetry. In the present paper, we identify the global causal geometry of these spacetimes and, in particular, establish that weakly regular Gowdy spacetimes are future causally geodesically complete. This result extends a theorem by Ringström for metrics with sufficiently high regularity. We emphasize that our proof of the energy decay is based on an energy functional inspired by the Gowdy-to-Ernst transformation. In order to establish the geodesic completeness property, we prove a higher regularity property concerning the metric coefficients along timelike curves and we provide a novel analysis of the geodesic equation for Gowdy spacetimes, which does not require high-order regularity estimates. Even when sufficient regularity is assumed, our proof provides an alternative and shorter proof of the energy decay and of the geodesic completeness property for Gowdy spacetimes.

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Hyperbolic Conservation Laws on Spacetimes

Cited by 8 publications

References 25 publications

Relativistic Burgers Equations on Curved Spacetimes. Derivation and Finite Volume Approximation

Relativistic Burgers Equations on Curved Spacetimes. Derivation and Finite Volume Approximation

A central-upwind geometry-preserving method for hyperbolic conservation laws on the sphere

Weakly regular T2-symmetric spacetimes. The future causal geometry of Gowdy spacetimes

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