arXiv subjects

Philippe G. LeFloch

Publications and source records attributed to Philippe G. LeFloch.

At least 91 records · Page 5Linked to original sources

arXiv2013

Numerical methods with controlled dissipation for small-scale dependent shocks

We provide a `user guide' to the literature of the past twenty years concerning the modeling and approximation of discontinuous solutions to nonlinear hyperbolic systems that admit small-scale dependent shock waves. We cover several classes of problems and solutions: nonclassical undercompressive shocks, hyperbolic systems in nonconservative form, boundary layer problems. We review the relevant models arising in continuum physics and describe the numerical methods that have been proposed to capture small-scale dependent solutions. In agreement with the general well-posedness theory, small-scale dependent solutions are characterized by a kinetic relation, a family of paths, or an admissible boundary set. We provide a review of numerical methods (front tracking schemes, finite difference schemes, finite volume schemes), which, at the discrete level, reproduce the effect of the physically-meaningful dissipation mechanisms of interest in the applications. An essential role is played by the equivalent equation associated with discrete schemes, which is found to be relevant even for solutions containing shock waves.

math.AP↗

arXiv2013

Quasilinear hyperbolic Fuchsian systems and AVTD behavior in T2-symmetric vacuum spacetimes

We set up the singular initial value problem for quasilinear hyperbolic Fuchsian systems of first order and establish an existence and uniqueness theory for this problem with smooth data and smooth coefficients (and with even lower regularity). We apply this theory in order to show the existence of smooth (generally not analytic) T2-symmetric solutions to the vacuum Einstein equations, which exhibit AVTD (asymptotically velocity term dominated) behavior in the neighborhood of their singularities and are polarized or half-polarized.

gr-qc↗

arXiv2012

The geometric finite volume method for compressible fluid flows on Schwarzschild spacetime

We consider the relativistic Euler equations governing spherically symmetric, perfect fluid flows on the outer domain of communication of Schwarzschild spacetime, and we introduce a version of the finite volume method which is formulated geometrically (without choosing coordinates a priori) and is well--balanced, in the sense that it preserves steady solutions to the Euler equations on the curved geometry under consideration. In order to formulate our method, we first derive a closed formula describing all steady and spherically symmetric solutions to the Euler equations posed on Schwarzschild spacetime. Second, we describe a finite volume method which is formulated geometrically from the family of steady solutions to the Euler system. Our scheme is second--order accurate and, as required, preserves the family of steady solutions at the discrete level. Numerical experiments are presented which demonstrate the efficiency and robustness of the proposed method even for solutions containing shock waves and nonlinear interacting wave patterns. As an application, we investigate the late--time asymptotics of perturbed steady solutions and demonstrate its convergence for late time toward another steady solution, taking the overall effect of the perturbation into account.

gr-qc↗

arXiv2012

Coupling techniques for nonlinear hyperbolic equations. III. The well-balanced approximation of thick interfaces

We continue our analysis of the coupling between nonlinear hyperbolic problems across possibly resonant interfaces. In the first two parts of this series, we introduced a new framework for coupling problems which is based on the so-called thin interface model and uses an augmented formulation and an additional unknown for the interface location; this framework has the advantage of avoiding any explicit modeling of the interface structure. In the present paper, we pursue our investigation of the augmented formulation and we introduce a new coupling framework which is now based on the so-called thick interface model. For scalar nonlinear hyperbolic equations in one space variable, we observe that the Cauchy problem is well-posed. Then, our main achievement in the present paper is the design of a new well-balanced finite volume scheme which is adapted to the thick interface model, together with a proof of its convergence toward the unique entropy solution (for a broad class of nonlinear hyperbolic equations). Due to the presence of a possibly resonant interface, the standard technique based on a total variation estimate does not apply, and DiPerna's uniqueness theorem must be used. Following a method proposed by Coquel and LeFloch, our proof relies on discrete entropy inequalities for the coupling problem and an estimate of the discrete entropy dissipation in the proposed scheme.

math.AP↗

arXiv2012

The finite energy method for compressible fluids. The Navier-Stokes-Korteweg model

This is the first of a series of papers devoted to the initial value problem for the Euler system of compressible fluids and augmented versions containing higher-order terms. We encompass solutions that have finite total energy and enjoy a certain symmetry (for instance, plane symmetry); these solutions may have unbounded amplitude and contain cavitation regions in which the mass density vanishes. In the present paper, we are interested in dispersive shock waves and analyze the zero viscosity-capillarity limit associated with the Navier-Stokes-Korteweg system. Specifically, we establish the existence of finite energy solutions as well as their convergence toward entropy solutions to the Euler system. We encompass a broad class of nonlinear Navier-Stokes-Korteweg constitutive laws, which is determined by two main conditions relating the viscosity and capillarity coefficients, that is, on one hand the strong coercivity condition (as we call it) which provides a favorable sign for the integrated dissipation associated with an effective energy, and on the other hand the tame capillarity condition (as we call it), which restricts pointwise the strength of the capillarity relatively to the viscosity. Rather mild conditions on the growth of the constitutive functions are aso imposed, which are required in order to define finite energy weak solutions to the Navier-Stokes-Korteweg system, even in the presence of cavitation. Our method of proof relies on fine algebraic properties of the Euler system and combines together energy and effective energy estimates, dissipation and effective dissipation estimates, a nonlinear Sobolev inequality, high-integrability properties for the mass density and for the velocity, and compactness properties based on entropies.

math.AP↗

arXiv2012

The linear stability of shock waves for the nonlinear Schrödinger-Inviscid Burgers system

We investigate the coupling between the nonlinear Schrödinger equation and the inviscid Burgers equation, a system which models interactions between short and long waves, for instance in fluids. Well-posedness for the associated Cauchy problem remains a difficult open problem, and we tackle it here via a linearization technique. Namely, we establish a linearized stability theorem for the Schrödinger--Burgers system, when the reference solution is an entropy--satisfying shock wave to Burgers equation. Our proof is based on suitable energy estimates and on properties of hyperbolic equations with discontinuous coefficients. Numerical experiments support and expand our theoretical results.

math.AP↗

arXiv2012

Weakly regular Einstein-Euler spacetimes with Gowdy symmetry. The global areal foliation

We consider weakly regular Gowdy-symmetric spacetimes on T3 satisfying the Einstein-Euler equations of general relativity, and we solve the initial value problem when the initial data set has bounded variation, only, so that the corresponding spacetime may contain impulsive gravitational waves as well as shock waves. By analyzing, both, future expanding and future contracting spacetimes, we establish the existence of a global foliation by spacelike hypersurfaces so that the time function coincides with the area of the surfaces of symmetry and asymptotically approaches infinity in the expanding case and zero in the contracting case. More precisely, the latter property in the contracting case holds provided the mass density does not exceed a certain threshold, which is a natural assumption since certain exceptional data with sufficiently large mass density are known to give rise to a Cauchy horizon, on which the area function attains a positive value. An earlier result by LeFloch and Rendall assumed a different class of weak regularity and did not determine the range of the area function in the contracting case. Our method of proof is based on a version of the random choice scheme adapted to the Einstein equations for the symmetry and regularity class under consideration. We also analyze the Einstein constraint equations under weak regularity.

gr-qc↗

arXiv2012

High-order asymptotic-preserving methods for fully nonlinear relaxation problems

We study solutions to nonlinear hyperbolic systems with fully nonlinear relaxation terms in the limit of, both, infinitely stiff relaxation and arbitrary late time. In this limit, the dynamics is governed by effective systems of parabolic type, with possibly degenerate and/or fully nonlinear diffusion terms. For this class of problems, we develop here an implicit-explicit method based on Runge-Kutta discretization in time, and we use this method in order to investigate several examples of interest in compressible fluid dynamics. Importantly, we impose here a realistic stability condition on the time-step and we demonstrate that solutions in the hyperbolic-to-parabolic regime can be computed numerically with high robustness and accuracy, even in presence of fully nonlinear relaxation terms.

math.AP↗

arXiv2012

Quasi-linear symmetric hyperbolic Fuchsian systems in several space dimensions

We establish existence and uniqueness results for the singular initial value problem associated with a class of quasilinear, symmetric hyperbolic, partial differential equations of Fuchsian type in several space dimensions. This is an extension of earlier work by the authors for the same problem in one space dimension.

gr-qc↗

arXiv2012

Relativistic Burgers equations on curved spacetimes. Derivation and finite volume approximation

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.

math.AP↗

arXiv2012

Coupling techniques for nonlinear hyperbolic equations. IV. Multi-component coupling and multidimensional well-balanced schemes

This series of papers is devoted to the formulation and the approximation of coupling problems for nonlinear hyperbolic equations. The coupling across an interface in the physical space is formulated in term of an augmented system of partial differential equations. In an earlier work, this strategy allowed us to develop a regularization method based on a thick interface model in one space variable. In the present paper, we significantly extend this framework and, in addition, encompass equations in several space variables. This new formulation includes the coupling of several distinct conservation laws and allows for a possible covering in space. Our main contributions are, on one hand, the design and analysis of a well-balanced finite volume method on general triangulations and, on the other hand, a proof of convergence of this method toward entropy solutions, extending Coquel, Cockburn, and LeFloch's theory (restricted to a single conservation law without coupling). The core of our analysis is, first, the derivation of entropy inequalities as well as a discrete entropy dissipation estimate and, second, a proof of convergence toward the entropy solution of the coupling problem.

math.AP↗

arXiv2012

Plane-symmetric spacetimes with positive cosmological constant. The case of stiff fluids

We consider plane-symmetric spacetimes satisfying Einstein's field equations with positive cosmological constant, when the matter is a fluid whose pressure is equal to its mass-energy density (i.e. a so-called stiff fluid). We study the initial-value problem for the associated Einstein equations and establish a global existence result. The late-time asymptotics of solutions is also rigorously derived, and we conclude that the spacetime approaches the de Sitter spacetime while the matter disperses asymptotically. A technical difficulty dealt with here lies in the fact that solutions may contain vacuum states as well as velocities approaching the speed of light, both possibilities leading to singular behavior in the evolution equations.

gr-qc↗

arXiv2011

New energy inequalities for tensorial wave equations on spacetimes that satisfy a one-sided bound

We consider several tensorial wave equations, specifically the equations of Maxwell, Yang-Mills, and Weyl fields, posed on a curved spacetime, and we establish new energy inequalities under certain one-sided geometric conditions. Our conditions restrict the lapse function and deformation tensor of the spacetime foliation, and turn out to be a one-sided and integral generalization of conditions recently proposed by Klainerman and Rodnianski as providing a continuation criterion for Einstein's field equations of general relativity. As we observe it here for the first time, one-sided conditions are sufficient to derive energy inequalities for certain tensorial equations, provided one takes advantage of some algebraic properties enjoyed by the natural energy functionals associated with the equations under consideration. Our method especially applies to the Bel-Robinson energy for Weyl fields, and our inequalities control the growth of the energy in a uniform way, with implied constants depending on the one-sided geometric bounds, only.

gr-qc↗

arXiv2011

Late-time/stiff relaxation asymptotic-preserving approximations of hyperbolic equations

We investigate the late-time asymptotic behavior of solutions to nonlinear hyperbolic systems of conservation laws containing stiff relaxation terms. First, we introduce a Chapman-Enskog-type asymptotic expansion and derive an effective system of equations describing the late-time/stiff relaxation singular limit. The structure of this new system is discussed and the role of a mathematical entropy is emphasized. Second, we propose a new finite volume discretization which, in late-time asymptotics, allows us to recover a discrete version of the same effective asymptotic system. This is achieved provided we suitably discretize the relaxation term in a way that depends on a matrix-valued free-parameter, chosen so that the desired asymptotic behavior is obtained. Our results are illustrated with several models of interest in continuum physics, and numerical experiments demonstrate the relevance of the proposed theory and numerical strategy.

math.AP↗

arXiv2011

Second-order hyperbolic Fuchsian systems. Asymptotic behavior of geodesics in Gowdy spacetimes

Recent work by the authors led to the development of a mathematical theory dealing with `second--order hyperbolic Fuchsian systems', as we call them. In the present paper, we adopt a physical standpoint and discuss the implications of this theory which provides one with a new tool to tackle the Einstein equations of general relativity (under certain symmetry assumptions). Specifically, we formulate the `Fuchsian singular initial value problem' and apply our general analysis to the broad class of vacuum Gowdy spacetimes with spatial toroidal topology. Our main focus is on providing a detailed description of the asymptotic geometry near the initial singularity of these inhomogeneous cosmological spacetimes and, especially, analyzing the asymptotic behavior of timelike geodesics ---which represent the trajectories of freely falling observers --- and null geodesics. In particular, we numerically construct Gowdy spacetimes which contain a black hole--like region together with a flat Minkowski--like region. By using the Fuchsian technique, we investigate the effect of the gravitational interaction between these two regions and we study the unexpected behavior of geodesic trajectories within the intermediate part of the spacetime limited by these two regions.

gr-qc↗

arXiv2011

Coupling techniques for nonlinear hyperbolic equations. I. Self-similar diffusion for thin interfaces

We investigate various analytical and numerical techniques for the coupling of nonlinear hyperbolic systems and, in particular, we introduce here an augmented formulation which allows for the modeling of the dynamics of interfaces between fluid flows. The main technical difficulty to be overcome lies in the possible resonance effect when wave speeds coincide and global hyperbolicity is lost. As a consequence, non-uniqueness of weak solutions is observed for the initial value problem which need to be supplemented with further admissibility conditions. This first paper is devoted to investigating these issues in the setting of self-similar vanishing viscosity approximations to the Riemann problem for general hyperbolic systems. Following earlier works by Joseph, LeFloch, and Tzavaras, we establish an existence theorem for the Riemann problem under fairly general structural assumptions on the nonlinear hyperbolic system and its regularization. Our main contribution consists of nonlinear wave interaction estimates for solutions which apply to resonant wave patterns.

math.AP↗

arXiv2011

A Godunov-type method for the shallow water equations with discontinuous topography in the resonant regime

We investigate the Riemann problem for the shallow water equations with variable and (possibly) discontinuous topography and provide a complete description of the properties of its solutions: existence; uniqueness in the non-resonant regime; multiple solutions in the resonant regime. This analysis leads us to a numerical algorithm that provides one with a Riemann solver. Next, we introduce a Godunov-type scheme based on this Riemann solver, which is well-balanced and of quasi-conservative form. Finally, we present numerical experiments which demonstrate the convergence of the proposed scheme even in the resonance regime, except in the limiting situation when Riemann data precisely belong to the resonance hypersurface.

math.AP↗

arXiv2011

Null injectivity estimate under an upper bound on the curvature

We establish a uniform estimate for the injectivity radius of the past null cone of a point in a general Lorentzian manifold foliated by spacelike hypersurfaces and satisfying an upper curvature bound. Precisely, our main assumptions are, on one hand, upper bounds on the null curvature of the spacetime and the lapse function of the foliation, and sup-norm bounds on the deformation tensors of the foliation. Our proof is inspired by techniques from Riemannian geometry, and it should be noted that we impose no restriction on the size of the curvature or deformation tensors, and allow for metrics that are "far" from the Minkowski one. The relevance of our estimate is illustrated with a class of plane-symmetric spacetimes which satisfy our assumptions but admit no uniform lower bound on the curvature not even in the L2 norm. The conditions we put forward, therefore, lead to a uniform control of the spacetime geometry and should be useful in the context of general relativity.

gr-qc↗