Strong Hyperbolicity Research Articles

Nonlinear studies of modifications to general relativity: Comparing different approaches

Studying the dynamical, nonlinear regime of modified theories of gravity remains a theoretical challenge that limits our ability to test general relativity. Here we consider two generally applicable, but approximate, methods for treating modifications to full general relativity that have been used to study binary black hole mergers and other phenomena in this regime, and compare solutions obtained by them to those from solving the full equations of motion. The first method evolves corrections to general relativity order by order in a perturbative expansion, while the second method introduces extra dynamical fields in such a way that strong hyperbolicity is recovered. We use shift-symmetric Einstein-scalar-Gauss-Bonnet gravity as a benchmark theory to illustrate the differences between these methods for several spacetimes of physical interest. We study the formation of scalar hair about initially nonspinning black holes, the collision of black holes with scalar charge, and the inspiral and merger of binary black holes. By directly comparing predictions, we assess the extent to which those from the approximate treatments can be meaningfully confronted with gravitational wave observations. We find that the order-by-order approach cannot faithfully track the solutions when the corrections to general relativity are non-negligible. The second approach, however, can provide consistent solutions, provided the timescale over which the dynamical fields are driven to their target values is made short compared to the physical timescales. Published by the American Physical Society 2024

High-Order ADER Discontinuous Galerkin Schemes for a Symmetric Hyperbolic Model of Compressible Barotropic Two-Fluid Flows

This paper presents a high-order discontinuous Galerkin (DG) finite-element method to solve the barotropic version of the conservative symmetric hyperbolic and thermodynamically compatible (SHTC) model of compressible two-phase flow, introduced by Romenski et al. in [59, 62], in multiple space dimensions. In the absence of algebraic source terms, the model is endowed with a curl constraint on the relative velocity field. In this paper, the hyperbolicity of the system is studied for the first time in the multidimensional case, showing that the original model is only weakly hyperbolic in multiple space dimensions. To restore the strong hyperbolicity, two different methodologies are used: (i) the explicit symmetrization of the system, which can be achieved by adding terms that contain linear combinations of the curl involution, similar to the Godunov-Powell terms in the MHD equations; (ii) the use of the hyperbolic generalized Lagrangian multiplier (GLM) curl-cleaning approach forwarded. The PDE system is solved using a high-order ADER-DG method with a posteriori subcell finite-volume limiter to deal with shock waves and the steep gradients in the volume fraction commonly appearing in the solutions of this type of model. To illustrate the performance of the method, several different test cases and benchmark problems have been run, showing the high order of the scheme and the good agreement when compared to reference solutions computed with other well-known methods.

Some recent advances on the method of moments in kinetic theory

This review presents some recent works on the construction of closure relations for moment systems extracted from a kinetic equation. A rough construction of those closures and the main properties of the resulting systems are described. Especially, based on the underlying kinetic equation, the main properties desired for such moment systems are the realizability, i.e. the positivity of an underlying kinetic solution, global strong hyperbolicity and entropy dissipation.

Causal, stable first-order viscous relativistic hydrodynamics with ideal gas microphysics

We present the first numerical analysis of causal, stable first-order relativistic hydrodynamics with ideal gas microphysics, based in the formalism developed by Bemfica, Disconzi, Noronha, and Kovtun (BDNK theory). The BDNK approach provides definitions for the conserved stress-energy tensor and baryon current, and rigorously proves causality, local well-posedness, strong hyperbolicity, and linear stability (about equilibrium) for the equations of motion, subject to a set of coupled nonlinear inequalities involving the undetermined model coefficients (the choice for which defines the ``hydrodynamic frame''). We present a class of hydrodynamic frames derived from the relativistic ideal gas ``gamma-law'' equation of state which satisfy the BDNK constraints, and explore the properties of the resulting model for a series of $(0+1)\mathrm{D}$ and $(1+1)\mathrm{D}$ tests in 4D Minkowski spacetime. These tests include a comparison of the dissipation mechanisms in Eckart, BDNK, and M\"uller-Israel-Stewart theories, as well as investigations of the impact of hydrodynamic frame on the causality and stability properties of Bjorken flow, planar shockwave, and heat flow solutions.

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.

Overdetermined ODEs and rigid periodic states in network dynamics

We consider four long-standing Rigidity Conjectures about synchrony and phase patterns for hyperbolic periodic orbits of admissible ODEs for networks. Proofs of stronger local versions of these conjectures, published in 2010–12, are now known to have a gap, but remain valid for a broad class of networks. Using different methods we prove local versions of the conjectures under a stronger condition, ‘strong hyperbolicity’, which is related to a network analogue of the Kupka–Smale Theorem. Under this condition we also deduce global versions of the conjectures and an analogue of the $H/K$ Theorem in equivariant dynamics.We prove the Rigidity Conjectures for all 1- and 2-colourings and all 2- and 3-node networks by showing that strong hyperbolicity is generic in these cases.

Hyperbolicity of First Order Quasi-Linear Equations

The theorem about equivalence of the strong hyperbolicity concept and the Friedrichs hyperbolicity concept for partial quasi-linear differential equations of the first order is proved. On the basis of this theorem, the necessary and sufficient conditions of hyperbolicity are found in terms of the matrix of the corresponding linearized first order equations system.

Lyapunov Exponents for Random Perturbations of Coupled Standard Maps

In this paper, we give a quantitative estimate for the sum of the first $N$ Lyapunov exponents for random perturbations of a natural class $2N$-dimensional volume-preserving systems exhibiting strong hyperbolicity on a large but noninvariant subset of phase space. Concrete models covered by our setting include systems of coupled standard maps, in both `weak' and `strong' coupling regimes.

On High Order ADER Discontinuous Galerkin Schemes for First Order Hyperbolic Reformulations of Nonlinear Dispersive Systems

This paper is on arbitrary high order fully discrete one-step ADER discontinuous Galerkin schemes with subcell finite volume limiters applied to a new class of first order hyperbolic reformulations of nonlinear dispersive systems based on an extended Lagrangian approach introduced by Dhaouadi et al. (Stud Appl Math 207:1–20, 2018), Favrie and Gavrilyuk (Nonlinearity 30:2718–2736, 2017). We consider the hyperbolic reformulations of two different nonlinear dispersive systems, namely the Serre–Green–Naghdi model of dispersive water waves and the defocusing nonlinear Schrödinger equation. The first order hyperbolic reformulation of the Schrödinger equation is endowed with a curl involution constraint that needs to be properly accounted for in multiple space dimensions. We show that the original model proposed in Dhaouadi et al. (2018) is only weakly hyperbolic in the multi-dimensional case and that strong hyperbolicity can be restored at the aid of a novel thermodynamically compatible GLM curl cleaning approach that accounts for the curl involution constraint in the PDE system. We show one and two-dimensional numerical results applied to both systems and compare them with available exact, numerical and experimental reference solutions whenever possible.

A note on causality conditions on covering spacetimes

A number of techniques in Lorentzian geometry, such as those used in the proofs of singularity theorems, depend on certain smooth coverings retaining interesting global geometric properties, including causal ones. In this note we give explicit examples showing that, unlike some of the more commonly adopted rungs of the causal ladder such as strong causality or global hyperbolicity, less-utilized conditions such as causal continuity or causal simplicity do not in general pass to coverings, as already speculated by one of the authors (EM). As a consequence, any result which relies on these causality conditions transferring to coverings must be revised accordingly. In particular, some amendments in the statement and proof of a version of the Gannon–Lee singularity theorem previously given by one of us (IPCS) are also presented here that address a gap in its original proof, simultaneously expanding its scope to spacetimes with lower causality.

High order ADER schemes and GLM curl cleaning for a first order hyperbolic formulation of compressible flow with surface tension

In this work, we introduce two novel reformulations of the weakly hyperbolic model for two-phase flow with surface tension, recently forwarded by Schmidmayer et al. In the model, the tracking of phase boundaries is achieved by using a new vector field, rather than a scalar tracer, so that the surface-force stress tensor can be expressed directly as an algebraic function of the state variables, without requiring the computation of gradients of the scalar tracer. An interesting and important feature of the model is that this interface field obeys a curl involution constraint, that is, the vector field is required to be curl-free at all times.The proposed modifications are intended to restore the strong hyperbolicity of the model, and are closely related to divergence-preserving numerical approaches developed in the field of numerical magnetohydrodynamics (MHD). The first strategy is based on the theory of Symmetric Hyperbolic and Thermodynamically Compatible (SHTC) systems forwarded by Godunov in the 60s and 70s and yields a modified system of governing equations which includes some symmetrisation terms, in analogy to the approach adopted later by Powell et al. in the 90s for the ideal MHD equations. The second technique is an extension of the hyperbolic Generalized Lagrangian Multiplier (GLM) divergence cleaning approach, forwarded by Munz et al. in applications to the Maxwell and MHD equations.We solve the resulting nonconservative hyperbolic partial differential equation (PDE) systems with high order ADER-WENO Finite Volume and ADER Discontinuous Galerkin (DG) methods with a posteriori Finite Volume subcell limiting and carry out a set of numerical tests concerning flows dominated by surface tension as well as shock-driven flows. We also provide a new exact solution to the equations, show convergence of the schemes for orders of accuracy up to ten in space and time, and investigate the role of hyperbolicity and of curl constraints in the long-term stability of the computations.

Strong hyperbolicity in Gevrey classes

We are concerned with the Cauchy problem in Gevrey classes and we are interested in finding condition on the principal symbol for which the Cauchy problem is stable under perturbation by lower order terms.

Hyperbolicity of general relativity in Bondi-like gauges

Bondi-like (single-null) characteristic formulations of general relativity are used for numerical work in both asymptotically flat and anti-de Sitter spacetimes. Well-posedness of the resulting systems of partial differential equations, however, remains an open question. The answer to this question affects accuracy, and potentially the reliability of conclusions drawn from numerical studies based on such formulations. A numerical approximation can converge to the continuum limit only for well-posed systems; for the initial value problem in the $L^2$ norm this is characterized by strong hyperbolicity. We find that, due to a shared pathological structure, the systems arising from the aforementioned formulations are however only weakly hyperbolic. We present numerical tests for toy models that demonstrate the consequence of this shortcoming in practice for the characteristic initial boundary value problem. Working with alternative norms in which our model problems may be well-posed we show that convergence may be recovered. Finally we examine well-posedness of a model for Cauchy-Characteristic-Matching in which model symmetric and weakly hyperbolic systems communicate through an interface, with the latter playing the role of GR in Bondi gauge on characteristic slices. We find that, due to the incompatibility of the norms associated with the two systems, the composite problem does not naturally admit energy estimates.

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.

Hamilton–Jacobi hydrodynamics of pulsating relativistic stars

The dynamics of self-gravitating fluid bodies is described by the Euler–Einstein system of partial differential equations. The break-down of well-posedness on the fluid–vacuum interface remains a challenging open problem, which is manifested in simulations of oscillating or inspiraling binary neutron-stars. We formulate and implement a well-posed canonical hydrodynamic scheme, suitable for neutron-star simulations in numerical general relativity. The scheme uses a variational principle by Carter–Lichnerowicz stating that barotropic fluid motions are conformally geodesic and Helmholtz’s third theorem stating that initially irrotational flows remain irrotational. We apply this scheme in 3 + 1 numerical general relativity to evolve the canonical momentum of a fluid element via the Hamilton–Jacobi equation. We explore a regularization scheme for the Euler equations, that uses a fiducial atmosphere in hydrostatic equilibrium and allows the pressure to vanish, while preserving strong hyperbolicity on the vacuum boundary. The new regularization scheme resolves a larger number of radial oscillation modes compared to standard, non-equilibrium atmosphere treatments.

A strong hyperbolicity property of locally symmetric varieties

We show that all subvarieties of a quotient of a bounded symmetric domain by a sufficiently small arithmetic discrete group of automorphisms are of general type. This result corresponds through the Green-Griffiths-Lang's conjecture to a well-known result of Nadel.

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.

Kobayashi hyperbolicity of the complements of general hypersurfaces of high degree

In this paper, we prove that in any projective manifold, the complements of general hypersurfaces of sufficiently large degree are Kobayashi hyperbolic. We also provide an effective lower bound on the degree. This confirms a conjecture by S. Kobayashi in 1970. Our proof, based on the theory of jet differentials, is obtained by reducing the problem to the construction of a particular example with strong hyperbolicity properties. This approach relies the construction of higher order logarithmic connections allowing us to construct logarithmic Wronskians. These logarithmic Wronskians are the building blocks of the more general logarithmic jet differentials we are able to construct. As a byproduct of our proof, we prove a more general result on the orbifold hyperbolicity for generic geometric orbifolds in the sense of Campana, with only one component and large multiplicities. We also establish a Second Main Theorem type result for holomorphic entire curves intersecting general hypersurfaces, and we prove the Kobayashi hyperbolicity of the cyclic cover of a general hypersurface, again with an explicit lower bound on the degree of all these hypersurfaces.

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.