On the well posedness of the Baumgarte-Shapiro-Shibata-Nakamura formulation of Einstein's field equations

Abstract

We give a well posed initial value formulation of the Baumgarte-Shapiro-Shibata-Nakamura form of Einstein's equations with gauge conditions given by a Bona-Masso like slicing condition for the lapse and a frozen shift. This is achieved by introducing extra variables and recasting the evolution equations into a first order symmetric hyperbolic system. We also consider the presence of artificial boundaries and derive a set of boundary conditions that guarantee that the resulting initial-boundary value problem is well posed, though not necessarily compatible with the constraints. In the case of dynamical gauge conditions for the lapse and shift we obtain a class of evolution equations which are strongly hyperbolic and so yield well posed initial value formulations.

References in corpus (31)

• On the Numerical Integration of Einstein's Field Equations
• Gauge conditions for long-term numerical black hole evolutions without excision
• New Formalism for Numerical Relativity
• Well-Posed Initial-Boundary Evolution in General Relativity
• Gravitational wave extraction and outer boundary conditions by perturbative matching
• First order hyperbolic formalism for Numerical Relativity
• Towards an understanding of the stability properties of the 3+1 evolution equations in general relativity
• Strongly hyperbolic second order Einstein's evolution equations
• Well posed constraint-preserving boundary conditions for the linearized Einstein equations
• Boundary conditions in linearized harmonic gravity
• Hyperbolicity of the BSSN system of Einstein evolution equations
• Constraint-preserving boundary conditions in numerical relativity
• Numerical treatment of the hyperboloidal initial value problem for the vacuum Einstein equations. II. The evolution equations
• Symmetric hyperbolicity and consistent boundary conditions for second-order Einstein equations
• Cauchy-perturbative matching and outer boundary conditions: computational studies
• Cauchy-perturbative matching and outer boundary conditions I: Methods and tests
• Novel finite-differencing techniques for numerical relativity: application to black hole excision
• Symmetric hyperbolic form of systems of second-order evolution equations subject to constraints
• Exploiting gauge and constraint freedom in hyperbolic formulations of Einstein's equations
• Convergence and stability in numerical relativity
• Dynamical shift conditions for the Z4 and BSSN hyperbolic formalisms
• A hyperbolic slicing condition adapted to Killing fields and densitized lapses
• Boundary conditions for hyperbolic formulations of the Einstein equations
• Einstein boundary conditions for the 3+1 Einstein equations
• Characteristic Evolution and Matching
• Detecting ill posed boundary conditions in General Relativity
• Impact of densitized lapse slicings on evolutions of a wobbling black hole
• Dynamical Gauge Conditions for the Einstein Evolution Equations
• Numerical stability of a new conformal-traceless 3+1 formulation of the Einstein equation
• Einstein boundary conditions in relation to constraint propagation for the initial-boundary value problem of the Einstein equations
• Hyperbolicity of the Kidder-Scheel-Teukolsky formulation of Einstein's equations coupled to a modified Bona-Masso slicing condition

Cited by in corpus (62)

• Numerical Relativity Using a Generalized Harmonic Decomposition
• Binary Neutron Star Mergers
• How to move a black hole without excision: gauge conditions for the numerical evolution of a moving puncture
• Turduckening black holes: an analytical and computational study
• Accurate black hole evolutions by fourth-order numerical relativity
• Continuum and Discrete Initial-Boundary-Value Problems and Einstein's Field Equations
• Error-analysis and comparison to analytical models of numerical waveforms produced by the NRAR Collaboration
• The Cauchy problem of scalar-tensor theories of gravity
• Covariant formulations of BSSN and the standard gauge
• Well-posedness of formulations of the Einstein equations with dynamical lapse and shift conditions
• On the final fate of compact boson star mergers
• A multi-block infrastructure for three-dimensional time-dependent numerical relativity
• Matched Filtering of Numerical Relativity Templates of Spinning Binary Black Holes
• Exploring New Physics Frontiers Through Numerical Relativity
• Excision without excision: the relativistic turducken
• Implementation of standard testbeds for numerical relativity
• Hyperbolicity of second-order in space systems of evolution equations
• Black hole head-on collisions and gravitational waves with fixed mesh-refinement and dynamic singularity excision
• Toward a dynamical shift condition for unequal mass black hole binary simulations
• Spherical symmetry as a test case for unconstrained hyperboloidal evolution
• A Reinvestigation of Moving Punctured Black Holes with a New Code
• Long term black hole evolution with the BSSN system by pseudospectral methods
• Comparing Fully General Relativistic and Newtonian Calculations of Structure Formation
• Numerical simulations with a first order BSSN formulation of Einstein's field equations
• BSSN in Spherical Symmetry
• Numerical stability for finite difference approximations of Einstein's equations
• Numerical relativity simulations of thick accretion disks around tilted Kerr black holes
• Discrete boundary treatment for the shifted wave equation
• A model problem for the initial-boundary value formulation of Einstein's field equations
• Nonlinear Evolution of Quadratic Gravity in 3+1 Dimensions
• Probing the puncture for black hole simulations
• Well-posedness of cubic Horndeski theories
• Geometrically motivated hyperbolic coordinate conditions for numerical relativity: Analysis, issues and implementations
• The Initial-Boundary Value Problem in General Relativity
• Absorbing boundary conditions for Einstein's field equations
• Boundary conditions for the Baumgarte-Shapiro-Shibata-Nakamura formulation of Einstein's field equations
• Stability of the puncture method with a generalized BSSN formulation
• Numerical modeling of black holes as sources of gravitational waves in a nutshell
• Free evolution of the hyperboloidal initial value problem in spherical symmetry
• Boundary Conditions for the Gravitational Field
• A spectral method algorithm for numerical simulations of gravitational fields
• Finite Difference Methods for Second Order in Space, First Order in Time Hyperbolic Systems and the Linear Shifted Wave Equation as a Model Problem in Numerical Relativity
• Constraint and gauge shocks in one-dimensional numerical relativity
• $\mathtt{bimEX}$: A Mathematica package for exact computations in $3+1$ bimetric relativity
• Oppenheimer-Snyder Collapse in Moving-Puncture Coordinates
• Covariant BSSN formulation in bimetric relativity
• A new first-order formulation of the Einstein equations exploiting analogies with electrodynamics
• On the Hyperbolicity and Stability of $3+1$ Formulations of Metric $f(R)$ Gravity
• Disembodied boundary data for Einstein's equations
• Assessing the stability of ultracompact spinning boson stars with nonlinear evolutions
• Hyperbolicity of High Order Systems of Evolution Equations
• Generalized Harmonic Equations in 3+1 Form
• The mean gauges in bimetric relativity
• The initial boundary value problem for free-evolution formulations of General Relativity
• Constraint preserving boundary conditions for the Baumgarte-Shapiro-Shibata-Nakamura formulation in spherical symmetry
• Numerical experiments of adjusted BSSN systems for controlling constraint violations
• Well-balanced high order finite difference WENO schemes for a first-order Z4 formulation of the Einstein field equations
• On constraint preservation and strong hyperbolicity
• A Note on the Strong Hyperbolicity of $f(R)$ Gravity with Dynamical Shifts
• A polytopal discrete de Rham scheme for the exterior calculus Einstein's equations
• Aliasing Instabilities in the Numerical Evolution of the Einstein Field Equations
• A spectral approach to numerical simulations of the ADM equations