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Manuel Tiglio

Publications and source records attributed to Manuel Tiglio.

53 records · Page 3Linked to original sources
arXiv2005

Multi-block simulations in general relativity: high order discretizations, numerical stability, and applications

The need to smoothly cover a computational domain of interest generically requires the adoption of several grids. To solve the problem of interest under this grid-structure one must ensure the suitable transfer of information among the different grids involved. In this work we discuss a technique that allows one to construct finite difference schemes of arbitrary high order which are guaranteed to satisfy linear numerical and strict stability. The technique relies on the use of difference operators satisfying summation by parts and {\it penalty techniques} to transfer information between the grids. This allows the derivation of semidiscrete energy estimates for problems admitting such estimates at the continuum. We analyze several aspects of this technique when used in conjuction with high order schemes and illustrate its use in one, two and three dimensional numerical relativity model problems with non-trivial topologies, including truly spherical black hole excision.

gr-qc↗
arXiv2005

Boundary conditions for Einstein's field equations: Analytical and numerical analysis

Outer boundary conditions for strongly and symmetric hyperbolic formulations of 3D Einstein's field equations with a live gauge condition are discussed. The boundary conditions have the property that they ensure constraint propagation and control in a sense made precise in this article the physical degrees of freedom at the boundary. We use Fourier-Laplace transformation techniques to find necessary conditions for the well posedness of the resulting initial-boundary value problem and integrate the resulting three-dimensional nonlinear equations using a finite-differencing code. We obtain a set of constraint-preserving boundary conditions which pass a robust numerical stability test. We explicitly compare these new boundary conditions to standard, maximally dissipative ones through Brill wave evolutions. Our numerical results explicitly show that in the latter case the constraint variables, describing the violation of the constraints, do not converge to zero when resolution is increased while for the new boundary conditions, the constraint variables do decrease as resolution is increased. As an application, we inject pulses of ``gravitational radiation'' through the boundaries of an initially flat spacetime domain, with enough amplitude to generate strong fields and induce large curvature scalars, showing that our boundary conditions are robust enough to handle nonlinear dynamics. We expect our boundary conditions to be useful for improving the accuracy and stability of current binary black hole and binary neutron star simulations, for a successful implementation of characteristic or perturbative matching techniques, and other applications. We also discuss limitations of our approach and possible future directions.

gr-qc↗
arXiv2004

Recent analytical and numerical techniques applied to the Einstein equations

Combining deeper insight of Einstein's equations with sophisticated numerical techniques promises the ability to construct accurate numerical implementations of these equations. We illustrate this in two examples, the numerical evolution of ``bubble'' and single black hole spacetimes. The former is chosen to demonstrate how accurate numerical solutions can answer open questions and even reveal unexpected phenomena. The latter illustrates some of the difficulties encountered in three-dimensional black hole simulations, and presents some possible remedies.

gr-qc↗
arXiv2004

Summation by parts and dissipation for domains with excised regions

We discuss finite difference techniques for hyperbolic equations in non-trivial domains, as those that arise when simulating black hole spacetimes. In particular, we construct dissipative and difference operators that satisfy the {\it summation by parts} property in domains with excised multiple cubic regions. This property can be used to derive semi-discrete energy estimates for the associated initial-boundary value problem which in turn can be used to prove numerical stability.

gr-qc↗
arXiv2004

3D simulations of Einstein's equations: symmetric hyperbolicity, live gauges and dynamic control of the constraints

We present three-dimensional simulations of Einstein equations implementing a symmetric hyperbolic system of equations with dynamical lapse. The numerical implementation makes use of techniques that guarantee linear numerical stability for the associated initial-boundary value problem. The code is first tested with a gauge wave solution, where rather larger amplitudes and for significantly longer times are obtained with respect to other state of the art implementations. Additionally, by minimizing a suitably defined energy for the constraints in terms of free constraint-functions in the formulation one can dynamically single out preferred values of these functions for the problem at hand. We apply the technique to fully three-dimensional simulations of a stationary black hole spacetime with excision of the singularity, considerably extending the lifetime of the simulations.

gr-qc↗
arXiv2004

The discrete energy method in numerical relativity: Towards long-term stability

The energy method can be used to identify well-posed initial boundary value problems for quasi-linear, symmetric hyperbolic partial differential equations with maximally dissipative boundary conditions. A similar analysis of the discrete system can be used to construct stable finite difference equations for these problems at the linear level. In this paper we apply these techniques to some test problems commonly used in numerical relativity and observe that while we obtain convergent schemes, fast growing modes, or ``artificial instabilities,'' contaminate the solution. We find that these growing modes can partially arise from the lack of a Leibnitz rule for discrete derivatives and discuss ways to limit this spurious growth.

gr-qc↗
arXiv2003

Dynamical control of the constraints growth in free evolutions of Einstein's equations

I present a new, simple method to dynamically control the growth of the discretized constraints during a free evolution of Einstein's equations. During an evolution, any given family of formulations is adjusted off the constraints surface in a way such that, for any chosen numerical method and arbitrary but fixed resolution, the constraints growth can be minimized with respect to the freedom allowed by the formulation. In particular, provided there is enough freedom, the discretized constraints can be maintained close to its initial truncation value for all times, or decay from it. No a priori knowledge of the solution is needed, and the method can be applied to any formulation of Einstein's equations without affecting hyperbolicity. This method is independent of the numerical algorithm and accounts for constraint violating modes introduced both by continuum instabilities of the formulation and by the numerical method.

gr-qc↗
arXiv2002

Well posed constraint-preserving boundary conditions for the linearized Einstein equations

In the Cauchy problem of general relativity one considers initial data that satisfies certain constraints. The evolution equations guarantee that the evolved variables will satisfy the constraints at later instants of time. This is only true within the domain of dependence of the initial data. If one wishes to consider situations where the evolutions are studied for longer intervals than the size of the domain of dependence, as is usually the case in three dimensional numerical relativity, one needs to give boundary data. The boundary data should be specified in such a way that the constraints are satisfied everywhere, at all times. In this paper we address this problem for the case of general relativity linearized around Minkowski space using the generalized Einstein-Christoffel symmetric hyperbolic system of evolution equations. We study the evolution equations for the constraints, specify boundary conditions for them that make them well posed and further choose these boundary conditions in such a way that the evolution equations for the metric variables are also well posed. We also consider the case of a manifold with a non-smooth boundary, as is the usual case of the cubic boxes commonly used in numerical relativity. The techniques discussed should be applicable to more general cases, as linearizations around more complicated backgrounds, and may be used to establish well posedness in the full non-linear case.

gr-qc↗
arXiv2002

Convergence and stability in numerical relativity

It is often the case in numerical relativity that schemes that are known to be convergent for well posed systems are used in evolutions of weakly hyperbolic (WH) formulations of Einstein's equations. Here we explicitly show that with several of the discretizations that have been used through out the years, this procedure leads to non-convergent schemes. That is, arbitrarily small initial errors are amplified without bound when resolution is increased, independently of the amount of numerical dissipation introduced. The lack of convergence introduced by this instability can be particularly subtle, in the sense that it can be missed by several convergence tests, especially in 3+1 dimensional codes. We propose tests and methods to analyze convergence that may help detect these situations.

gr-qc↗
arXiv2002

Exploiting gauge and constraint freedom in hyperbolic formulations of Einstein's equations

We present new many-parameter families of strongly and symmetric hyperbolic formulations of Einstein's equations that include quite general algebraic and live gauge conditions for the lapse. The first system that we present has 30 variables and incorporates an algebraic relationship between the lapse and the determinant of the three metric that generalizes the densitized lapse prescription. The second system has 34 variables and uses a family of live gauges that generalizes the Bona-Masso slicing conditions. These systems have free parameters even after imposing hyperbolicity and are expected to be useful in 3D numerical evolutions. We discuss under what conditions there are no superluminal characteristic speeds.

gr-qc↗
arXiv2002

Stability properties of a formulation of Einstein's equations

We study the stability properties of the Kidder-Scheel-Teukolsky (KST) many-parameter formulation of Einstein's equations for weak gravitational waves on flat space-time from a continuum and numerical point of view. At the continuum, performing a linearized analysis of the equations around flat spacetime, it turns out that they have, essentially, no non-principal terms. As a consequence, in the weak field limit the stability properties of this formulation depend only on the level of hyperbolicity of the system. At the discrete level we present some simple one-dimensional simulations using the KST family. The goal is to analyze the type of instabilities that appear as one changes parameter values in the formulation. Lessons learnt in this analysis can be applied in other formulations with similar properties.

gr-qc↗
arXiv2002

Hyperbolicity of the BSSN system of Einstein evolution equations

We discuss an equivalence between the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation of the Einstein evolution equations, a subfamiliy of the Kidder--Scheel--Teukolsky formulation, and other strongly or symmetric hyperbolic first order systems with fixed shift and densitized lapse. This allows us to show under which conditions the BSSN system is, in a sense to be discussed, hyperbolic. This desirable property may account in part for the empirically observed better behavior of the BSSN formulation in numerical evolutions involving black holes.

gr-qc↗
arXiv2001

Constraint-preserving boundary conditions in numerical relativity

This is the first paper in a series aimed to implement boundary conditions consistent with the constraints' propagation in 3D numerical relativity. Here we consider spherically symmetric black hole spacetimes in vacuum or with a minimally coupled scalar field, within the Einstein-Christoffel symmetric hyperbolic formulation of Einstein's equations. By exploiting the characteristic propagation of the main variables and constraints, we are able to single out the only free modes at the outer boundary for these problems. In the vacuum case a single free modes exists which corresponds to a gauge freedom, while in the matter case an extra mode exists which is associated with the scalar field. We make use of the fact that the EC formulation has no superluminal characteristic speeds to excise the singularity. We present a second-order, finite difference discretization to treat these scenarios, where we implement these constraint-preserving boundary conditions, and are able to evolve the system for essentially unlimited times. As a test of the robustness of our approach, we allow large pulses of gauge and scalar field enter the domain through the outer boundary. We reproduce expected results, such as trivial (in the physical sense) evolution in the vacuum case (even in gauge- dynamical simulations), and the tail decay for the scalar field.

gr-qc↗
arXiv2001

Close limit evolution of Kerr-Schild type initial data for binary black holes

We evolve the binary black hole initial data family proposed by Bishop {\em et al.} in the limit in which the black holes are close to each other. We present an exact solution of the linearized initial value problem based on their proposal and make use of a recently introduced generalized formalism for studying perturbations of Schwarzschild black holes in arbitrary coordinates to perform the evolution. We clarify the meaning of the free parameters of the initial data family through the results for the radiated energy and waveforms from the black hole collision.

gr-qc↗
arXiv2001

Gauge invariant perturbations of Schwarzschild black holes in horizon-penetrating coordinates

We derive a geometrical version of the Regge-Wheeler and Zerilli equations, which allows us to study gravitational perturbations on an arbitrary spherically symmetric slicing of a Schwarzschild black hole. We explain how to obtain the gauge-invariant part of the metric perturbations from the amplitudes obeying our generalized Regge-Wheeler and Zerilli equations and vice-versa. We also give a general expression for the radiated energy at infinity, and establish the relation between our geometrical equations and the Teukolsky formalism. The results presented in this paper are expected to be useful for the close-limit approximation to black hole collisions, for the Cauchy perturbative matching problem, and for the study of isolated horizons.

gr-qc↗
arXiv2001

A cure for unstable numerical evolutions of single black holes: adjusting the standard ADM equations

Numerical codes based on a direct implementation of the standard ADM formulation of Einstein's equations have generally failed to provide long-term stable and convergent evolutions of black hole spacetimes when excision is used to remove the singularities. We show that, for the case of a single black hole in spherical symmetry, it is possible to circumvent these problems by adding to the evolution equations terms involving the constraints, thus adjusting the standard ADM system. We investigate the effect that the choice of the lapse and shift has on the stability properties of numerical simulations and thus on the form of the added constraint term. To facilitate this task, we introduce the concept of quasi well-posedness, a version of well-posedness suitable for ADM-like systems involving second-order spatial derivatives.

gr-qc↗