Strongly Hyperbolic Research Articles

On constraint preservation and strong hyperbolicity

We use partial differential equations (PDEs) to describe physical systems. In general, these equations include evolution and constraint equations. One method used to find solutions to these equations is the free-evolution approach, which consists in obtaining the solutions of the entire system by solving only the evolution equations. Certainly, this is valid only when the chosen initial data satisfies the constraints and the constraints are preserved in the evolution. In this paper, we establish the sufficient conditions required for the PDEs of the system to guarantee the constraint preservation. This is achieved by considering quasi-linear first-order PDEs, assuming the sufficient condition and deriving strongly hyperbolic first-order partial differential evolution equations for the constraints. We show that, in general, these constraint evolution equations correspond to a family of equations parametrized by a set of free parameters. We also explain how these parameters fix the propagation velocities of the constraints. As application examples of this framework, we study the constraint conservation of the Maxwell electrodynamics and the wave equations in arbitrary space–times. We conclude that the constraint evolution equations are unique in the Maxwell case and a family in the wave equation case.

On necessary and sufficient conditions for strong hyperbolicity in systems with constraints

In this work, we study constant-coefficient first order systems of partial differential equations and give necessary and sufficient conditions for those systems to have a well-posed Cauchy problem. In many physical applications, due to the presence of constraints, the number of equations in the PDE system is larger than the number of unknowns, thus the standard Kreiss conditions can not be directly applied to check whether the system admits a well-posed initial value formulation. In this work, we find necessary and sufficient conditions such that there exists a reduced set of equations, of the same dimensionality as the set of unknowns, which satisfy Kreiss conditions and so are well defined and properly behaved evolution equations. We do that by studying the systems using the Kronecker decomposition of matrix pencils and, once the conditions are met, finding specific families of reductions which render the system strongly hyperbolic. We show the power of the theory in some examples: Klein Gordon, the ADM, and the BSSN equations by writing them as first order systems, and studying their Kronecker decomposition and general reductions.

Covariant BSSN formulation in bimetric relativity

Numerical integration of the field equations in bimetric relativity is necessary to obtain solutions describing realistic systems. Thus, it is crucial to recast the equations as a well-posed problem. In general relativity, under certain assumptions, the covariant BSSN formulation is a strongly hyperbolic formulation of the Einstein equations, hence its Cauchy problem is well-posed. In this paper, we establish the covariant BSSN formulation of the bimetric field equations. It shares many features with the corresponding formulation in general relativity, but there are a few fundamental differences between them. Some of these differences depend on the gauge choice and alter the hyperbolic structure of the system of partial differential equations compared to general relativity. Accordingly, the strong hyperbolicity of the system cannot be claimed yet, under the same assumptions as in general relativity. In the paper, we stress the differences compared with general relativity and state the main issues that should be tackled next, to draw a roadmap towards numerical bimetric relativity.

Two applications of strong hyperbolicity

We present two analytic applications of the fact that a hyperbolic group can be endowed with a strongly hyperbolic metric. The first application concerns the crossed product C∗-algebra defined by the action of a hyperbolic group on its boundary. We construct a natural time flow, involving the Busemann cocycle on the boundary. This flow has a natural KMS state, coming from the Hausdorff measure on the boundary, which is furthermore unique when the group is torsion-free. The second application is a short new proof of the fact that a hyperbolic group admits a proper isometric action on an ℓp-space for large enough p.

A necessary condition ensuring the strong hyperbolicity of first-order systems

We study strong hyperbolicity of first-order partial differential equations for systems with differential constraints. In these cases, the number of equations is larger than the unknown fields, therefore, the standard Kreiss necessary and sufficient conditions of strong hyperbolicity do not directly apply. To deal with this problem, one introduces a new tensor, called a reduction, which selects a subset of equations with the aim of using them as evolution equations for the unknown. If that tensor leads to a strongly hyperbolic system we call it a hyperbolizer. There might exist many of them or none. A question arises on whether a given system admits any hyperbolization at all. To sort-out this issue, we look for a condition on the system, such that, if it is satisfied, there is no hyperbolic reduction. To that purpose we look at the singular value decomposition of the whole system and study certain one parameter families ([Formula: see text]) of perturbations of the principal symbol. We look for the perturbed singular values around the vanishing ones and show that if they behave as [Formula: see text], with [Formula: see text], then there does not exist any hyperbolizer. In addition, we further notice that the validity or failure of this condition can be established in a simple and invariant way. Finally, we apply the theory to examples in physics, such as Force-Free Electrodynamics in Euler potentials form and charged fluids with finite conductivity. We find that they do not admit any hyperbolization.

Conformal and covariant Z4 formulation of the Einstein equations: Strongly hyperbolic first-order reduction and solution with discontinuous Galerkin schemes

We present a strongly hyperbolic first-order formulation of the Einstein equations based on the conformal and covariant Z4 system (CCZ4) with constraint-violation damping, which we refer to as FO-CCZ4. As CCZ4, this formulation combines the advantages of a conformal and traceless formulation, with the suppression of constraint violations given by the damping terms, but being first order in time and space, it is particularly suited for a discontinuous Galerkin (DG) implementation. The strongly hyperbolic first-order formulation has been obtained by making careful use of first and second-order ordering constraints. A proof of strong hyperbolicity is given for a selected choice of standard gauges via an analytical computation of the entire eigenstructure of the FO-CCZ4 system. The resulting governing partial differential equations system is written in non-conservative form and requires the evolution of 58 unknowns. A key feature of our formulation is that the first-order CCZ4 system decouples into a set of pure ordinary differential equations and a reduced hyperbolic system of partial differential equations that contains only linearly degenerate fields. We implement FO-CCZ4 in a high-order path-conservative arbitrary-high-order-method-using-derivatives (ADER)-DG scheme with adaptive mesh refinement and local time-stepping, supplemented with a third-order ADER-WENO subcell finite-volume limiter in order to deal with singularities arising with black holes. We validate the correctness of the formulation through a series of standard tests in vacuum, performed in one, two and three spatial dimensions, and also present preliminary results on the evolution of binary black-hole systems. To the best of our knowledge, these are the first successful three-dimensional simulations of moving punctures carried out with high-order DG schemes using a first-order formulation of the Einstein equations.

On the local well-posedness of Lovelock and Horndeski theories

We investigate local well-posedness of the initial value problem for Lovelock and Horndeski theories of gravity. A necessary condition for local well-posedness is strong hyperbolicity of the equations of motion. Even weak hyperbolicity can fail for strong fields so we restrict to weak fields. The Einstein equation is known to be strongly hyperbolic in harmonic gauge so we study Lovelock theories in harmonic gauge. We show that the equation of motion is always weakly hyperbolic for weak fields but, in a generic weak-field background, it is not strongly hyperbolic. For Horndeski theories, we prove that, for weak fields, the equation of motion is always weakly hyperbolic in any generalized harmonic gauge. For some Horndeski theories there exists a generalized harmonic gauge for which the equation of motion is strongly hyperbolic in a weak-field background. This includes "k-essence" like theories. However, for more general Horndeski theories, there is no generalized harmonic gauge for which the equation of motion is strongly hyperbolic in a generic weak-field background. Our results show that the standard method used to establish local well-posedness of the Einstein equation does not extend to Lovelock or general Horndeski theories. This raises the possibility that these theories may not admit a well-posed initial value problem even for weak fields.

Strong hyperbolicity

We propose the metric notion of strong hyperbolicity as a way of obtaining hyperbolicity with sharp additional properties. Specically, strongly hyperbolic spaces are Gromov hyperbolic spaces that are metrically well-behaved at infinity, and, under weak geodesic assumptions, they are strongly bolic as well. We show that CAT(–1) spaces are strongly hyperbolic. On the way, we determine the best constant of hyperbolicity for the standard hyperbolic plane \mathbb H^2 . We also show that the Green metric defined by a random walk on a hyperbolic group is strongly hyperbolic. A measure-theoretic consequence at the boundary is that the harmonic measure defined by a random walk is a visual Hausdorff measure.

Hyperbolicity of high-order systems of evolution equations

We study properties of evolution equations which are first order in time and arbitrary order in space (FTNS). Following Gundlach and Martín-García (2006) we define strong and symmetric hyperbolicity for FTNS systems and examine the relationship between these definitions, and the analogous concepts for first-order systems. We demonstrate equivalence of the FTNS definition of strong hyperbolicity with the existence of a strongly hyperbolic first-order reduction. We also demonstrate equivalence of the FTNS definition, up to N = 4, of symmetric hyperbolicity with the existence of a symmetric-hyperbolic first-order reduction.

AN INTRODUCTION TO WELL-POSEDNESS AND FREE-EVOLUTION

These lecture notes accompany two classes given at the NRHEP2 school. In the first lecture I introduce the basic concepts used for analyzing well-posedness, that is the existence of a unique solution depending continuously on given data, of evolution partial differential equations. I show how strong hyperbolicity guarantees well-posedness of the initial value problem. Symmetric hyperbolic systems are shown to render the initial boundary value problem well-posed with maximally dissipative boundary conditions. I discuss the Laplace–Fourier method for analyzing the initial boundary value problem. Finally, I state how these notions extend to systems that are first-order in time and second-order in space. In the second lecture I discuss the effect that the gauge freedom of electromagnetism has on the PDE status of the initial value problem. I focus on gauge choices, strong-hyperbolicity and the construction of constraint preserving boundary conditions. I show that strongly hyperbolic pure gauges can be used to build strongly hyperbolic formulations. I examine which of these formulations is additionally symmetric hyperbolic and finally demonstrate that the system can be made boundary stable.

Hyperbolicity of second order in space systems of evolution equations

A possible definition of strong/symmetric hyperbolicity for a second-order system of evolution equations is that it admits a reduction to first order which is strongly/symmetric hyperbolic. We investigate the general system that admits a reduction to first order and give necessary and sufficient criteria for strong/symmetric hyperbolicity of the reduction in terms of the principal part of the original second-order system. An alternative definition of strong hyperbolicity is based on the existence of a complete set of characteristic variables, and an alternative definition of symmetric hyperbolicity is based on the existence of a conserved (up to lower-order terms) energy. Both these definitions are made without any explicit reduction. Finally, strong hyperbolicity can be defined through a pseudo-differential reduction to first order. We prove that both definitions of symmetric hyperbolicity are equivalent and that all three definitions of strong hyperbolicity are equivalent (in three space dimensions). We show how to impose maximally dissipative boundary conditions on any symmetric hyperbolic second-order system. We prove that if the second-order system is strongly hyperbolic, any closed constraint evolution system associated with it is also strongly hyperbolic, and that the characteristic variables of the constraint system are derivatives of a subset of the characteristic variables of the main system, with the same speeds.

Complex strongly hyperbolic [formula omitted] systems, reduced dimension and hermitian systems

a is a complex matrix valued 4 × 4 strongly hyperbolic operator; we state the following result: if the reduced real dimension of a is superior or equal to 13, a is hermitian in a convenient basis. We consider a matrix valued 4 × 4 differential operator, the coefficients of which are complex a ( D ) = I D 0 + a ( D ′ ) ; the real reduced dimension d ( a ) of the operator is the real dimension of the vector space of the matrices: { ξ 0 I + ∑ 1 ⩽ k ⩽ n ( Re a k + i Im a k ) ξ k ; ( ξ 0 , ξ ′ ) = ( ξ 0 , ξ 1 , … , ξ n ) ∈ R n + 1 } . We state the following theorem: if the reduced dimension: d ( a ) ⩾ 4 2 − 3 = 13 , if a ( D ) is strongly hyperbolic, then there exist a complex invertible matrix T, such that T −1 a ( ξ ) T is hermitian, ∀ ξ. The corresponding results for real coefficients and any m was studied in [J. Vaillant, Ann. Scuola Norm. Sup. Pisa Cl. Sc. 5 (1978) 405; Y. Oshime, J. Math. Kyoto Univ. 31 (1991) 937; T. Nishitani, Ann. Scuola Norm. Sup. Pisa Cl. Sc. 21 (1994) 97; J. Vaillant, Ann. Scuola Norm. Sup. Pisa Cl. Sc. 4 (XXIX) (2000) 839; J. Vaillant, Bull. Soc. Roy. Sc. Liege 70 (4–6) (2001) 407; J. Vaillant, in: Proc. 3rd International ISAAC Congress, World Scientific, 2003, pp. 1073–1080; J. Vaillant, in: Partial Diff. Equations and Math. Physics, in memory of Jean Leray, Birkhäuser, 2003, pp. 195, 223]. In the case of complex coefficients, if we replace the assumption of strong hyperbolicity by the assumption of real diagonalizability, if m ⩾ 3 , if d ( a ) ⩾ m 2 − 2 , the theorems were stated in [Y. Oshime, Publ. Res. Inst. Math. Sc. Kyoto Univ. 28 (1992) 223; J. Vaillant, in: Hyperbolic and Related Topics, International Press, 2003, pp. 403–422; J. Vaillant, Stud. Math. Bulgar. 15 (2003) 133]. In the case of real coefficients, the results were extended to the case of variable coefficients [T. Nishitani, J. Vaillant, Tsukuba J. Math. 25 (2001) 165; ibid. 27 (2003) 389]; the authors prepare the extension theorem to the case of complex variable coefficients.

STRONGLY HYPERBOLIC SYSTEMS IN GENERAL RELATIVITY

We discuss several topics related to the notion of strong hyperbolicity which are of interest in general relativity. After introducing the concept and showing its relevance we provide some covariant definitions of strong hyperbolicity. We then prove that if a system is strongly hyperbolic with respect to a given hypersurface, then it is also strongly hyperbolic with respect to any nearby surface. We then study for how much these hypersurfaces can be deformed and discuss then causality, namely what the maximal propagation speed in any given direction is. In contrast with the symmetric hyperbolic case, for which the proof of causality is geometrical and direct, relaying in energy estimates, the proof for general strongly hyperbolic systems is indirect for it is based in Holmgren's theorem. To show that the concept is needed in the area of general relativity we discuss two results for which the theory of symmetric hyperbolic systems shows to be insufficient. The first deals with the hyperbolicity analysis of systems which are second order in space derivatives; they include certain versions of the ADM and the BSSN families of equations. This analysis is considerably simplified by introducing pseudo-differential first-order evolution equations. Well-posedness for some members of the latter family systems is established by showing they satisfy the strong hyperbolicity property. Furthermore it is shown that many other systems of such families are only weakly hyperbolic, implying they should not be used for numerical modeling. The second result deals with systems having constraints. The question posed is which hyperbolicity properties, if any, are inherited from the original evolution system by the subsidiary system satisfied by the constraint quantities. The answer is that, subject to some condition on the constraints, if the evolution system is strongly hyperbolic then the subsidiary system is also strongly hyperbolic and the causality properties of both are identical.

Mixed Problem for Hyperbolic Systems of First Order

In the last ten years, a general theory of mixed problems for linear hyperbolic systems of first order has been developed. In this paper, we treat the L2 well-posed mixed problems for strongly hyperbolic systems of first order with constant coefficients. It is well known that Cauchy problem for hyperbolic systems of first order with constant coefficients is L2 well-posed if and only if it is strongly hyperbolic DL2j. The family of strongly hyperbolic systems contains the strictly and symmetric hyperbolic systems. But, Strang's condition for strong hyper bolicity seems to be useless except for Cauchy problem. So, in Sec. 3, we prove another condition for strong hyperbolicity which is useful for not only Cauchy problem, but also mixed problems, the lacunas of Riemann's matrix of strongly hyperbolic systems [1], the propagation of singularities Q8] and others. The L2 well-posed mixed problems for the strictly and symmetric hyperbolic systems have been fully investigated. But, even if hyperbolic systems have constant coefficients, nothing is known in general for ones with multiple characteristics except for systems with constant multiple characteristics [[KT] and 2x2 systems [JL3]. We consider the mixed problem for hyperbolic systems of first order with constant coefficients: