arXiv subjects

David Hilditch

Publications and source records attributed to David Hilditch.

75 records · Page 5Linked to original sources

arXiv2010

Hyperbolic formulations of General Relativity with Hamiltonian structure

With the aim of deriving symmetric hyperbolic free-evolution systems for GR that possess Hamiltonian structure and allow for the popular puncture gauge condition we analyze the hyperbolicity of Hamiltonian systems. We develop helpful tools which are applicable to either the first order in time, second order in space or the fully second order form of the equations of motion. For toy models we find that the Hamiltonian structure can simplify the proof of symmetric hyperbolicity. In GR we use a special structure of the principal part to prove symmetric hyperbolicity of a formulation that includes gauge conditions which are very similar to the puncture gauge.

gr-qc↗

arXiv2007

Comments on Bona-Masso type slicing conditions in long-term black hole evolutions

We review in generality why time-independent endstates can be reached in black hole and collapse simulations, with and without excision. We characterise the Killing states of the Bona-Masso slicing condition with time derivative along the normals to the slice ("BMn") as solutions of a mixed elliptic/hyperbolic differential equation on the slice. We show numerically that these steady states can be reached as end states from typical initial data with excision but can be reached with the puncture method only if the puncture is not numerically well resolved. During the evolution, BMn slicings often form gauge shocks. It may be that these are not seen in current 3D simulations only through lack of resolution, although we expect that they can be avoided with some care. Finally we point out that excision with BMn as currently implemented is ill-posed and therefore not expected to converge; this can be cured. In technical appendixes, we derive the equations of pure gauge systems on a fixed spacetime, and bring the BSSN/NOR equations into 3-dimensional tensor form suitable for multiple coordinate patches or spherical polar coordinates.

gr-qc↗

arXiv2006

Asymptotically null slices in numerical relativity: mathematical analysis and spherical wave equation tests

We investigate the use of asymptotically null slices combined with stretching or compactification of the radial coordinate for the numerical simulation of asymptotically flat spacetimes. We consider a 1-parameter family of coordinates characterised by the asymptotic relation $r\sim R^{1-n}$ between the physical radius $R$ and coordinate radius $r$, and the asymptotic relation $K\sim R^{n/2-1}$ for the extrinsic curvature of the slices. These slices are asymptotically null in the sense that their Lorentz factor relative to stationary observers diverges as $Γ\sim R^{n/2}$. While $1<n\le 2$ slices intersect $\scri$, $0< n\le 1$ slices end at $i^0$. We carry out numerical tests with the spherical wave equation on Minkowski and Schwarzschild spacetime. Simulations using our coordinates with $0<n\le 2$ achieve higher accuracy at lower computational cost in following outgoing waves to very large radius than using standard $n=0$ slices without compactification. Power-law tails in Schwarzschild are also correctly represented.

gr-qc↗