Structure-Preserving Shock-Capturing Methods: Late-Time Asymptotics, Curved Geometry, Small-Scale Dissipation, and Nonconservative Products

LeFloch, Philippe G.

doi:10.1007/978-3-319-02839-2_4

Cited by 7 publications

(6 citation statements)

References 59 publications

(71 reference statements)

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“…It is outside the scope of this short Note to explain this notion here, and we refer the reader to the textbook 13 and the lecture notes 12,14 and the historical references therein. Importantly, based on these scattering maps we need to design structurepreserving algorithms, which are typically of front tracking type, or of shock capturing type with well-controlled dissipation 15 . The algorithms are designed in order to mimic analytical properties at the discrete level and this requirement may take very different flavors: divergence form, spacelike decay, timelike decay shock-capturing, energy balance laws, asymptotics on singularities, etc.…”

Section: Regime Of Small Viscosity and Capillaritymentioning

confidence: 99%

Gravitational singularities, scattering maps for bouncing, and structure-preserving algorithms

LeFloch¹

2021

Preprint

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This note emphasizes the role of multi-scale wave structures and junction conditions in many fields of physics, from the dynamics of fluids with non-convex equations of state to the study of gravitational singularities and bouncing cosmologies in general relativity. Concerning the definition and construction of bouncing spacetimes, we review the recent proposal in collaboration with B. Le Floch and G. Veneziano based on the notion of singularity scattering maps. We also present recent numerical investigations of smallscale phenomena arising in compressible fluid flows on FRLW or Kasner geometries for which we developed structure-preserving algorithms.

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Section: Regime Of Small Viscosity and Capillaritymentioning

confidence: 99%

Gravitational singularities, scattering maps for bouncing, and structure-preserving algorithms

LeFloch¹

2021

Preprint

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“…The inviscid Burgers equation is an important model in computational fluid dynamics and provides the simplest (yet challenging) example of a nonlinear hyperbolic conservation law. Recently, a relativistic generalization of the standard Burgers equation was introduced on curved spacetimes and studied by LeFloch and collaborators [1][2][3][4][9][10][11][12]. This relativistic Burgers equation, as it is now called, takes into account geometrical effects and satisfies the Lorentz invariance property, also enjoyed by the Euler equations of relativistic compressible fluids.…”

Section: Aim Of This Papermentioning

confidence: 99%

A Finite Volume Method for the Relativistic Burgers Equation on a FLRW Background Spacetime

Ceylan¹,

LeFloch²,

Okutmustur³

2018

CICP

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A relativistic generalization of the inviscid Burgers equation was introduced by LeFloch and co-authors and was recently investigated numerically on a Schwarzschild background. We extend this analysis to a Friedmann-Lemaître-Robertson-Walker (FLRW) background, which is more challenging due to the existence of time-dependent, spatially homogeneous solutions. We present a derivation of the model of interest and we study its basic properties, including the class of spatially homogeneous solutions. Then, we design a second-order accurate scheme based on the finite volume methodology, which provides us with a tool for investigating the properties of solutions. Computational experiments demonstrate the efficiency of the proposed scheme for numerically capturing weak solutions.

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“…For the theory of weak solutions to hyperbolic conservation laws on geometric background, we refer to [1,3,4,5] and [9,11,13,15,20]. The numerical computation for such equations has also recently received some attention; see for instance [16, ?, 19].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic structure of cosmological fluid flows in one and two space dimensions: a numerical study

Cao,

Ghazizadeh,

LeFloch

2019

Preprint

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We consider an isothermal compressible fluid evolving on a cosmological background which may be either expanding or contracting toward the future. The Euler equations governing such a flow consist of two nonlinear hyperbolic balance laws which we treat in one and in two space dimensions. We design a finite volume scheme which is fourth-order accurate in time and second-order accurate in space. This scheme allows us to compute weak solutions containing shock waves and, by design, is well-balanced in the sense that it preserves exactly a special class of solutions. Using this scheme, we investigate the asymptotic structure of the fluid when the time variable approaches infinity (in the expanding regime) or approaches zero (in the contracting regime). We study these two limits by introducing a suitable rescaling of the density and velocity variables and, in turn, we analyze the effects induced by the geometric terms (of expanding or contracting nature) and the nonlinear interactions between shocks. Extensive numerical experiments in one and in two space dimensions are performed in order to support our observations.

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Structure-Preserving Shock-Capturing Methods: Late-Time Asymptotics, Curved Geometry, Small-Scale Dissipation, and Nonconservative Products

Abstract: Given the averaged values u − K along e − K and u − K e 0 along e 0 ∈ ∂ 0 K, we need an approximation u + K of the solution u along e + K . The second term in (40) can be approximated by

Cited by 7 publications

References 59 publications

Gravitational singularities, scattering maps for bouncing, and structure-preserving algorithms

Gravitational singularities, scattering maps for bouncing, and structure-preserving algorithms

A Finite Volume Method for the Relativistic Burgers Equation on a FLRW Background Spacetime

Asymptotic structure of cosmological fluid flows in one and two space dimensions: a numerical study

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