Constraint-preserving Boundary Conditions Research Articles

Novel scheme for simulating the force-free equations: Boundary conditions and the evolution of solutions towards stationarity

Force-Free Electrodynamics (FFE) describes a particular regime of magnetically dominated relativistic plasmas, which arises on several astrophysical scenarios of interest such as pulsars or active galactic nuclei. In this article, we present a full 3D numerical implementation of the FFE evolution around a Kerr black hole. The novelty of our approach is three-folded: i) We use the "multi-block" technique to represent a domain with $S^2 \times \mathbb{R}^{+}$ topology within a stable finite-differences scheme. ii) We employ as evolution equations those arising from a covariant hyperbolization of the FFE system. iii) We implement stable and constraint-preserving boundary conditions to represent an outer region given by a uniform magnetic field aligned or misaligned respect to the symmetry axis. We find stationary jet solutions which reach equilibrium --through boundary conditions-- with the outer numerical surface. This is so, even when the outer boundary is located very close to the central region (i.e. $r_{out}\sim 10M $). These numerical solutions reproduce most of the known results for analogue astrophysical settings.

Pseudospectral method for gravitational wave collapse

We present a new pseudospectral code, bamps, for numerical relativity written with the evolution of collapsing gravitational waves in mind. We employ the first order generalized harmonic gauge formulation. The relevant theory is reviewed and the numerical method is critically examined and specialized for the task at hand. In particular we investigate formulation parameters, gauge and constraint preserving boundary conditions well-suited to non-vanishing gauge source functions. Different types of axisymmetric twist-free moment of time symmetry gravitational wave initial data are discussed. A treatment of the axisymmetric apparent horizon condition is presented with careful attention to regularity on axis. Our apparent horizon finder is then evaluated in a number of test cases. Moving on to evolutions, we investigate modifications to the generalized harmonic gauge constraint damping scheme to improve conservation in the strong field regime. We demonstrate strong-scaling of our pseudospectral penalty code. We employ the Cartoon method to efficiently evolve axisymmetric data in our 3+1 dimensional code. We perform test evolutions of Schwarzschild perturbed by gravitational waves and by gauge pulses, both to demonstrate the use of our blackhole excision scheme and for comparison with earlier results. Finally numerical evolutions of supercritical Brill waves are presented to demonstrate durability of the excision scheme for the dynamical formation of a blackhole.

Constraint preserving boundary conditions for the Baumgarte–Shapiro–Shibata–Nakamura formulation in spherical symmetry

We introduce a set of constraint preserving boundary conditions for the Baumgarte–Shapiro–Shibata–Nakamura formulation of the Einstein evolution equations in spherical symmetry, based on its hyperbolic structure. While the outgoing eigenfields are left to propagate freely off the numerical grid, boundary conditions are set to enforce that the incoming eigenfields donʼt introduce spurious reflections and, more importantly, that there are no fields introduced at the boundary that violate the constraint equations. In order to do this we adopt two different approaches to set boundary conditions for the extrinsic curvature, by expressing either the radial or the time derivative of its associated ingoing eigenfield in terms of the constraints. We find that these boundary conditions are very robust in practice, allowing us to perform long lasting evolutions that remain accurate and stable, and that converge to a solution that satisfies the constraints all the way to the boundary.

ON BOUNDARY CONDITIONS FOR FIRST-ORDER SYMMETRIC HYPERBOLIC SYSTEMS WITH CONSTRAINTS

The Cauchy problem for many first-order symmetric hyperbolic (FOSH) systems is constraint preserving, i.e. the solution satisfies certain spatial differential constraints whenever the initial data does. Frequently, artificial space cut-offs are performed for such evolution systems, usually out of the necessity for finite computational domains. However, it may easily happen that boundary conditions at the artificial boundary for such a system lead to an initial boundary value problem which, while well-posed, does not preserve the constraints. Here we consider the problem of finding constraint-preserving boundary conditions for constrained FOSH systems in the well-posed class of maximal non-negative boundary conditions. Based on a characterization of maximal non-negative boundary conditions, we discuss a systematic technique for finding such boundary conditions that preserve the constraints, pending that the constraints satisfy a FOSH system themselves. We exemplify this technique by analyzing a system of wave equations in a first-order formulation subject to divergence constraints.

Compact binary evolutions with the Z4c formulation

Numerical relativity simulations of compact binaries with the Z4c and BSSNOK formulations are compared. The Z4c formulation is advantageous in every case considered. In simulations of non-vacuum spacetimes the constraint violations due to truncation errors are between one and three orders of magnitude lower in the Z4c evolutions. Improvements are also found in the accuracy of the computed gravitational radiation. For equal-mass irrotational binary neutron star evolutions we find that the absolute errors in phase and amplitude of the waveforms can be up to a factor of four smaller. The quality of the Z4c numerical data is also demonstrated by a remarkably accurate computation of the ADM mass from surface integrals. For equal-mass non-spinning binary puncture black hole evolutions we find that the absolute errors in phase and amplitude of the waveforms can be up to a factor of two smaller. In the same evolutions we find that away from the punctures the Hamiltonian constraint violation is reduced by between one and two orders of magnitude. Furthermore, the utility of gravitational radiation controlling, constraint preserving boundary conditions for the Z4c formulation is demonstrated. The evolution of spacetimes containing a single compact object confirm earlier results in spherical symmetry. The boundary conditions avoid spurious and non-convergent effects present in high resolution runs with either formulation with a more naive boundary treatment. We conclude that Z4c is preferable to BSSNOK for the numerical solution of the 3+1 Einstein equations with the puncture gauge.

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.

Constraint preserving boundary conditions for the Z4c formulation of general relativity

We discuss high order absorbing constraint preserving boundary conditions for the Z4c formulation of general relativity coupled to the moving puncture family of gauges. We are primarily concerned with the constraint preservation and absorption properties of these conditions. In the frozen coefficient approximation, with an appropriate first order pseudo-differential reduction, we show that the constraint subsystem is boundary stable on a four dimensional compact manifold. We analyze the remainder of the initial boundary value problem for a spherical reduction of the Z4c formulation with a particular choice of the puncture gauge. Numerical evidence for the efficacy of the conditions is presented in spherical symmetry.

Constraint-preserving boundary conditions in the3+1first-order approach

A set of energy-momentum constraint-preserving boundary conditions is proposed for the first-order Z4 case. The stability of a simple numerical implementation is tested in the linear regime (robust stability test), both with the standard corner and vertex treatment and with a modified finite-differences stencil for boundary points which avoids corners and vertices even in Cartesian-like grids. Moreover, the proposed boundary conditions are tested in a strong-field scenario, the Gowdy waves metric, showing the expected rate of convergence. The accumulated amount of energy-momentum constraint violations is similar or even smaller than the one generated by either periodic or reflection conditions, which are exact in the Gowdy waves case. As a side theoretical result, a new symmetrizer is explicitly given, which extends the parametric domain of symmetric hyperbolicity for the Z4 formalism. The application of these results to first-order Baumgarte-Shapiro-Shibata-Nakamura--like formalisms is also considered.

Gowdy waves as a test-bed for constraint-preserving boundary conditions

Gowdy waves, one of the standard 'apples with apples' tests, is proposed as a test-bed for constraint-preserving boundary conditions in the non-linear regime. As an illustration, energy-constraint preservation is separately tested in the Z4 framework. Both algebraic conditions, derived from energy estimates, and derivative conditions, deduced from the constraint-propagation system, are considered. The numerical errors at the boundary are of the same order than those at the interior points.

Disembodied boundary data for Einstein’s equations

A strongly well-posed initial boundary value problem based upon constraint-preserving boundary conditions of the Sommerfeld type has been established for the harmonic formulation of the vacuum Einstein's equations. These Sommerfeld conditions have been previously presented in a four-dimensional geometric form. Here we recast the associated boundary data as three-dimensional tensor fields intrinsic to the boundary. This provides a geometric presentation of the boundary data analogous to the three-dimensional presentation of Cauchy data in terms of three-metric and extrinsic curvature. In particular, diffeomorphisms of the boundary data lead to vacuum spacetimes with isometric geometries. The proof of well-posedness is valid for the harmonic formulation and its generalizations. The Sommerfeld conditions can be directly applied to existing harmonic codes which have been used in simulating binary black holes, thus ensuring boundary stability of the underlying analytic system. The geometric form of the boundary conditions also allows them to be formally applied to any metric formulation of Einstein's equations, although well-posedness of the boundary problem is no longer ensured. We discuss to what extent such a formal application might be implemented in a constraint-preserving manner to $3+1$ formulations, such as the Baumgarte-Shapiro-Shibata-Nakamura system which has been highly successful in binary black hole simulation.

Well-posed constraint-preserving boundary conditions for the AA formulation of Einstein's equations

In this paper, we address the problem of artificial boundary conditions for the symmetric linearized hyperbolic formulation of Einstein's equations introduced in [A.M. Alekseenko, D.N. Arnold, New first-order formulation for the Einstein equations, Phys. Rev. D (3) 68 (6) (2003) 064013, 6 pp.]. We prescribe simple boundary conditions that make the problem well-posed and preserve the constraints.

Geometrization of metric boundary data for Einstein’s equations

The principle part of Einstein equations in the harmonic gauge consists of a constrained system of 10 curved space wave equations for the components of the space-time metric. A well-posed initial boundary value problem based upon a new formulation of constraint-preserving boundary conditions of the Sommerfeld type has recently been established for such systems. In this paper these boundary conditions are recast in a geometric form. This serves as a first step toward their application to other metric formulations of Einstein's equations.

Implementation of higher-order absorbing boundary conditions for the Einstein equations

We present an implementation of absorbing boundary conditions for the Einstein equations based on the recent work of Buchman and Sarbach. In this paper, we assume that spacetime may be linearized about Minkowski space close to the outer boundary, which is taken to be a coordinate sphere. We reformulate the boundary conditions as conditions on the gauge-invariant Regge–Wheeler–Zerilli scalars. Higher-order radial derivatives are eliminated by rewriting the boundary conditions as a system of ODEs for a set of auxiliary variables intrinsic to the boundary. From these we construct boundary data for a set of well-posed constraint-preserving boundary conditions for the Einstein equations in a first-order generalized harmonic formulation. This construction has direct applications to outer boundary conditions in simulations of isolated systems (e.g., binary black holes) as well as to the problem of Cauchy-perturbative matching. As a test problem for our numerical implementation, we consider linearized multipolar gravitational waves in TT gauge, with angular momentum numbers ℓ = 2 (Teukolsky waves), 3 and 4. We demonstrate that the perfectly absorbing boundary condition of order L = ℓ yields no spurious reflections to linear order in perturbation theory. This is in contrast to the lower-order absorbing boundary conditions with L < ℓ, which include the widely used freezing-Ψ0 boundary condition that imposes the vanishing of the Newman–Penrose scalar Ψ0.

Constraint-preserving boundary treatment for a harmonic formulation of the Einstein equations

We present a set of well-posed constraint-preserving boundary conditions for a first-order in time, second-order in space, harmonic formulation of the Einstein equations. The boundary conditions are tested using robust stability, linear and nonlinear waves, and are found to be both less reflective and constraint preserving than standard Sommerfeld-type boundary conditions.

Constraint preserving boundary conditions for the Ideal Newtonian MHD equations

We study and develop constraint preserving boundary conditions for the Newtonian magnetohydrodynamic equations and analyze the behavior of the numerical solution upon considering different possible options. We concentrate on both the standard ideal MHD system and the one augmented by a “pseudo potential” to control the divergence free constraint. We show how the boundary conditions developed significantly reduce the violations generated at the boundaries at the numerical level and how lessen their influence in the interior of the computational domain by making use of the available freedom in the equations.

Constraint-Preserving Boundary Conditions for the Linearized Baumgarte-Shapiro-Shibata-Nakamura Formulation

We derive two sets of explicit homogeneous algebraic constraint-preserving boundary conditions for the second-order in time reduction of the linearized Baumgarte-Shapiro-Shibata-Nakamura (BSSN) system. Our second-order reduction involves components of the linearized extrinsic curvature only. An initial-boundary value problem for the original linearized BSSN system is formulated and the existence of the solution is proved using the properties of the reduced system. A treatment is proposed for the full nonlinear BSSN system to construct constraint-preserving boundary conditions without invoking the second order in time reduction. Energy estimates on the principal part of the BSSN system (which is first order in temporal and second order in spatial derivatives) are obtained. Generalizations to the case of nonhomogeneous boundary data are proposed.

WELL-POSED INITIAL-BOUNDARY VALUE PROBLEM FOR A CONSTRAINED EVOLUTION SYSTEM AND RADIATION-CONTROLLING CONSTRAINT-PRESERVING BOUNDARY CONDITIONS

A well-posed initial-boundary value problem is formulated for the model problem of the vector wave equation subject to the divergence-free constraint. Existence, uniqueness and stability of the solution is proved by reduction to a system evolving the constraint quantity statically, i.e. the second time derivative of the constraint quantity is zero. A new set of radiation-controlling constraint-preserving boundary conditions is constructed for the free evolution problem. Comparison between the new conditions and the standard constraint-preserving boundary conditions is made using the Fourier–Laplace analysis and the power series decomposition in time. The new boundary conditions satisfy the Kreiss condition and are free from the ill-posed modes growing polynomially in time.

Testing outer boundary treatments for the Einstein equations

Various methods of treating outer boundaries in numerical relativity are compared using a simple test problem: a Schwarzschild black hole with an outgoing gravitational wave perturbation. Numerical solutions computed using different boundary treatments are compared to a ‘reference’ numerical solution obtained by placing the outer boundary at a very large radius. For each boundary treatment, the full solutions including constraint violations and extracted gravitational waves are compared to those of the reference solution, thereby assessing the reflections caused by the artificial boundary. These tests are based on a first-order generalized harmonic formulation of the Einstein equations and are implemented using a pseudo-spectral collocation method. Constraint-preserving boundary conditions for this system are reviewed, and an improved boundary condition on the gauge degrees of freedom is presented. Alternate boundary conditions evaluated here include freezing the incoming characteristic fields, Sommerfeld boundary conditions, and the constraint-preserving boundary conditions of Kreiss and Winicour. Rather different approaches to boundary treatments, such as sponge layers and spatial compactification, are also tested. Overall the best treatment found here combines boundary conditions that preserve the constraints, freeze the Newman–Penrose scalar Ψ0, and control gauge reflections.

Constraint-preserving Sommerfeld conditions for the harmonic Einstein equations

The principle part of Einstein equations in the harmonic gauge consists of a constrained system of 10 curved space wave equations for the components of the space-time metric. A new formulation of constraint-preserving boundary conditions of the Sommerfeld type for such systems has recently been proposed. We implement these boundary conditions in a nonlinear 3D evolution code and test their accuracy.

Towards absorbing outer boundaries in general relativity

We construct exact solutions to the Bianchi equations on a flat spacetime background. When the constraints are satisfied, these solutions represent in- and outgoing linearized gravitational radiation. We then consider the Bianchi equations on a subset of flat spacetime of the form [0, T] × BR, where BR is a ball of radius R, and analyse different kinds of boundary conditions on ∂BR. Our main results are as follows. (i) We give an explicit analytic example showing that boundary conditions obtained from freezing the incoming characteristic fields to their initial values are not compatible with the constraints. (ii) With the help of the exact solutions constructed, we determine the amount of artificial reflection of gravitational radiation from constraint-preserving boundary conditions which freeze the Weyl scalar Ψ0 to its initial value. For monochromatic radiation with wave number k and arbitrary angular momentum number ℓ ⩾ 2, the amount of reflection decays as (kR)−4 for large kR. (iii) For each L ⩾ 2, we construct new local constraint-preserving boundary conditions which perfectly absorb linearized radiation with ℓ ⩽ L. (iv) We generalize our analysis to a weakly curved background of mass M and compute first-order corrections in M/R to the reflection coefficients for quadrupolar odd-parity radiation. For our new boundary condition with L = 2, the reflection coefficient is smaller than that for the freezing Ψ0 boundary condition by a factor of M/R for kR > 1.04. Implications of these results for numerical simulations of binary black holes on finite domains are discussed.