Supplement to Summation by Parts, Projections, and Stability. I

Olsson, Pelle

doi:10.2307/2153512

Cited by 19 publications

(13 citation statements)

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“…Estimates for the term involving the commutator [ D, A ] have been given in [317, 319, 318, 407]. In order to discuss them, we first notice that the SBP property of D with respect to Σ and the symmetry of A imply that the operator B := [ D, A ] is symmetric with respect to Σ , for all grid functions u and v .…”

Section: Spatial Approximations: Finite Differencesmentioning

confidence: 99%

Continuum and Discrete Initial-Boundary Value Problems and Einstein’s Field Equations

Sarbach

Tiglio

2012

Living Rev. Relativ.

150

168

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Many evolution problems in physics are described by partial differential equations on an infinite domain; therefore, one is interested in the solutions to such problems for a given initial dataset. A prominent example is the binary black-hole problem within Einstein’s theory of gravitation, in which one computes the gravitational radiation emitted from the inspiral of the two black holes, merger and ringdown. Powerful mathematical tools can be used to establish qualitative statements about the solutions, such as their existence, uniqueness, continuous dependence on the initial data, or their asymptotic behavior over large time scales. However, one is often interested in computing the solution itself, and unless the partial differential equation is very simple, or the initial data possesses a high degree of symmetry, this computation requires approximation by numerical discretization. When solving such discrete problems on a machine, one is faced with a finite limit to computational resources, which leads to the replacement of the infinite continuum domain with a finite computer grid. This, in turn, leads to a discrete initial-boundary value problem. The hope is to recover, with high accuracy, the exact solution in the limit where the grid spacing converges to zero with the boundary being pushed to infinity.The goal of this article is to review some of the theory necessary to understand the continuum and discrete initial boundary-value problems arising from hyperbolic partial differential equations and to discuss its applications to numerical relativity; in particular, we present well-posed initial and initial-boundary value formulations of Einstein’s equations, and we discuss multi-domain high-order finite difference and spectral methods to solve them.

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Section: Spatial Approximations: Finite Differencesmentioning

confidence: 99%

Continuum and Discrete Initial-Boundary Value Problems and Einstein’s Field Equations

Sarbach

Tiglio

2012

Living Rev. Relativ.

150

168

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“…In the system here considered, the constraints are added to the RHS of Eqs. (1)(2)(3)(4)(5), and the spacetime constraintfunctions {γ, ζ, η, χ, ξ} are introduced as multiplicative factors to the constraints. Requiring the evolution system to be symmetric hyperbolic imposes algebraic conditions on these factors, as discussed below, and they are not treated as completely independent.…”

Section: The Symmetric Hyperbolic Formulation Usedmentioning

confidence: 99%

“…That is, by constructing a numerically stable scheme for the IBVP under consideration. One way of doing so is by constructing difference operators and imposing the discrete boundary conditions in a way such that the steps followed at the continuum to show well posedness can be reproduced at the discrete level [4,5,6].…”

Section: Introductionmentioning

confidence: 99%

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

2004

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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 constraintfunctions 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.

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“…Impose boundary conditions via orthogonal projections [3]. This ensures the consistent treatment of the boundaries, guaranteeing the correct handling of modes propagating towards, from and tangential to the boundaries.…”

Section: Guidelines For a Stable Numerical Implementationmentioning

confidence: 99%

Recent Analytical and Numerical Techniques Applied to the Einstein Equations

Neilsen

Lehner

Sarbach

et al.

Analytical and Numerical Approaches to Mathematical Relativity

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Summary. 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.

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Supplement to Summation by Parts, Projections, and Stability. I

Cited by 19 publications

References 0 publications

Continuum and Discrete Initial-Boundary Value Problems and Einstein’s Field Equations

Continuum and Discrete Initial-Boundary Value Problems and Einstein’s Field Equations

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

Recent Analytical and Numerical Techniques Applied to the Einstein Equations

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