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Generalized harmonic formulation in spherical symmetry
General Relativity and Quantum Cosmology · Submitted 18 Aug 2009
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Surprisingly, we find that many of the drivers that perform well in 3+1 calculations that use Cartesian coordinates, are considerably less effective in spherical symmetry, where some of them are, in fact, unstable.
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The abstract, sentence by sentence
In this pedagogically structured article, we describe a generalized harmonic formulation of the Einstein equations in spherical symmetry which is regular at the origin. The generalized harmonic approach has attracted significant attention in numerical relativity over the past few years, especially as applied to the problem of binary inspiral and merger. A key issue when using the technique is the choice of the gauge source functions, and recent work has provided several prescriptions for gauge drivers designed to evolve these functions in a controlled way. We numerically investigate the parameter spaces of some of these drivers in the context of fully non-linear collapse of a real, massless scalar field, and determine nearly optimal parameter settings for specific situations. Surprisingly, we find that many of the drivers that perform well in 3+1 calculations that use Cartesian coordinates, are considerably less effective in spherical symmetry, where some of them are, in fact, unstable.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this pedagogically structured article, we describe a generalized harmonic formulation of the Einstein equations in spherical symmetry which is regular at the origin. | contribution 0.99 | 0.84 | 0.12 | asserted 0.99 |
| s1 | The generalized harmonic approach has attracted significant attention in numerical relativity over the past few years, especially as applied to the problem of binary inspiral and merger. | background 1.00 | 0.04 | 0.02 | asserted 1.00 |
| s2 | A key issue when using the technique is the choice of the gauge source functions, and recent work has provided several prescriptions for gauge drivers designed to evolve these functions in a controlled way. | background 1.00 | 0.04 | 0.03 | asserted 1.00 |
| s3 | We numerically investigate the parameter spaces of some of these drivers in the context of fully non-linear collapse of a real, massless scalar field, and determine nearly optimal parameter settings for specific situations. | contribution 0.81 | 0.96 | 0.12 | asserted 1.00 |
| s4 | Surprisingly, we find that many of the drivers that perform well in 3+1 calculations that use Cartesian coordinates, are considerably less effective in spherical symmetry, where some of them are, in fact, unstable. | result 1.00 | 0.96 | 0.63 | asserted 1.00 |
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The paper
Original abstract
In this pedagogically structured article, we describe a generalized harmonic formulation of the Einstein equations in spherical symmetry which is regular at the origin. The generalized harmonic approach has attracted significant attention in numerical relativity over the past few years, especially as applied to the problem of binary inspiral and merger. A key issue when using the technique is the choice of the gauge source functions, and recent work has provided several prescriptions for gauge drivers designed to evolve these functions in a controlled way. We numerically investigate the parameter spaces of some of these drivers in the context of fully non-linear collapse of a real, massless scalar field, and determine nearly optimal parameter settings for specific situations. Surprisingly, we find that many of the drivers that perform well in 3+1 calculations that use Cartesian coordinates, are considerably less effective in spherical symmetry, where some of them are, in fact, unstable.
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References
How the literature uses it
7 arXiv papers cite this in 17 sentences
Papers citing it · 8 sentences without a confident label
Builds on it 1 paper
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Applications of the GH approach in spherically-symmetric situations were studied in [ref] and here we employ the method in axisymmetry.
An axisymmetric generalized harmonic evolution code · Introduction · 0.94
Uses its method or tool 3 papers
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Hence, instead of introducing $$ we opted to implement a more straightforward regularization method that maintains $S$ as a fundamental dynamical variable and uses analytical Taylor-series expansion to compute its value near the axis; see [ref] for more details.
An axisymmetric generalized harmonic evolution code · Dimensionally reduced equations and boundary conditions · 0.80
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This allows using compactification meaningfully [ref] .
On critical collapse of gravitational waves · The equations and a method of their solution · 0.80
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For instance, for 4th order the resulting expression is [equation] This procedure is mentioned in section 5.2 of [ref] .
Free evolution of the hyperboloidal initial value problem in spherical symmetry · Non-staggered grid · 0.95
Compares with it 1 paper
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On the other hand, our experiments with various drivers—described in detail in Sec. [ref]—reveal that in the case of symmetries these drivers are considerably less effective relative to the $3+1$ simulations that use Cartesian coordinates, a conclusion similar to that drawn in [ref] .
An axisymmetric generalized harmonic evolution code · Conclusion · 0.84
Cites it as background 3 papers
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Variants of gauge drivers were further investigated in the recent [ref] .
An axisymmetric generalized harmonic evolution code · Coordinate conditions · 0.91
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In particular we note the previously mentioned pioneering works concerned with the non-linear development of the Gregory-Laflamme instability [ref] of cosmic strings [ref] ; studies of gravitational collapse [ref] ; static situations [ref] ; and the alternative approach, based on the Cartoon method [ref] , that has been developed and tested by Yoshino and Shibata [ref] .
New frontiers in Numerical Relativity · Higher-dimensional gravity · 0.96
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Some pioneering works have been concerned with the non-linear development of the Gregory-Laflamme instability [ref] of cosmic strings [ref] and gravitational collapse, with spherical symmetry [ref] , axial symmetry [ref] or even static situations [ref] .
Numerical relativity for D dimensional axially symmetric space-times: formalism and code tests · Space-times with symmetries · 0.93
What it cites · 42 sentences without a confident label
Uses the method or tool of 7 papers
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Again following [ref] we write the driver condition as
Gauge Drivers for the Generalized Harmonic Einstein Equations · Shift conditions · 0.94
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In our implementation we follow Pretorius [ref] by adding constraint damping terms in a fashion inspired by studies of the so-called $$-systems [ref] .
Einstein's Equations with Asymptotically Stable Constraint Propagation · Generalized harmonic formulation · 1.00
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In our implementation we follow Pretorius [ref] by adding constraint damping terms in a fashion inspired by studies of the so-called $$-systems [ref] .
Numerical Relativity Using a Generalized Harmonic Decomposition · Generalized harmonic formulation · 1.00
Compares with 4 papers
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Our studies of evolutions using several coordinate drivers lead us to conclude that, in spherical geometries, the gauge drivers discussed in [ref] are less effective relative to the $3+1$ simulations that use Cartesian coordinates, and it would be very interesting to understand this issue in more detail.
Gauge Drivers for the Generalized Harmonic Einstein Equations · Conclusions · 0.94
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Our studies of evolutions using several coordinate drivers lead us to conclude that, in spherical geometries, the gauge drivers discussed in [ref] are less effective relative to the $3+1$ simulations that use Cartesian coordinates, and it would be very interesting to understand this issue in more detail.
High-accuracy waveforms for binary black hole inspiral, merger, and ringdown · Conclusions · 0.94
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Our studies of evolutions using several coordinate drivers lead us to conclude that, in spherical geometries, the gauge drivers discussed in [ref] are less effective relative to the $3+1$ simulations that use Cartesian coordinates, and it would be very interesting to understand this issue in more detail.
Evolution of Binary Black Hole Spacetimes · Conclusions · 0.94
Cites as background 10 papers
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The $Γ$-driver strategy is related to the minimal distortion condition [ref] which is designed to minimize the time variation of $_ij$ (see e.g. [ref] ).
Gauge conditions for long-term numerical black hole evolutions without excision · Shift conditions · 0.94
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The $Γ$-driver strategy is related to the minimal distortion condition [ref] which is designed to minimize the time variation of $_ij$ (see e.g. [ref] ).
Numerical Relativity and Compact Binaries · Shift conditions · 0.94
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One can view the generalized harmonic (GH) formulation of general relativity as a particular method that eliminates the mixed second derivatives appearing in ([ref]) [ref] .
Numerical Relativity Using a Generalized Harmonic Decomposition · Generalized harmonic formulation · 0.87
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