7 research outputs found
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.Comment: 18 pages, 22 figure
Hamiltonian Relaxation
Due to the complexity of the required numerical codes, many of the new
formulations for the evolution of the gravitational fields in numerical
relativity are not tested on binary evolutions. We introduce in this paper a
new testing ground for numerical methods based on the simulation of binary
neutron stars. This numerical setup is used to develop a new technique, the
Hamiltonian relaxation (HR), that is benchmarked against the currently most
stable simulations based on the BSSN method. We show that, while the length of
the HR run is somewhat shorter than the equivalent BSSN simulation, the HR
technique improves the overall quality of the simulation, not only regarding
the satisfaction of the Hamiltonian constraint, but also the behavior of the
total angular momentum of the binary. The latest quantity agrees well with
post-Newtonian estimations for point-mass binaries in circular orbits.Comment: More detailed description of the numerical implementation added and
some typos corrected. Version accepted for publication in Class. and Quantum
Gravit
Relativistic MHD and black hole excision: Formulation and initial tests
A new algorithm for solving the general relativistic MHD equations is
described in this paper. We design our scheme to incorporate black hole
excision with smooth boundaries, and to simplify solving the combined Einstein
and MHD equations with AMR. The fluid equations are solved using a finite
difference Convex ENO method. Excision is implemented using overlapping grids.
Elliptic and hyperbolic divergence cleaning techniques allow for maximum
flexibility in choosing coordinate systems, and we compare both methods for a
standard problem. Numerical results of standard test problems are presented in
two-dimensional flat space using excision, overlapping grids, and elliptic and
hyperbolic divergence cleaning.Comment: 22 pages, 8 figure
Numerical Relativity: A review
Computer simulations are enabling researchers to investigate systems which
are extremely difficult to handle analytically. In the particular case of
General Relativity, numerical models have proved extremely valuable for
investigations of strong field scenarios and been crucial to reveal unexpected
phenomena. Considerable efforts are being spent to simulate astrophysically
relevant simulations, understand different aspects of the theory and even
provide insights in the search for a quantum theory of gravity. In the present
article I review the present status of the field of Numerical Relativity,
describe the techniques most commonly used and discuss open problems and (some)
future prospects.Comment: 2 References added; 1 corrected. 67 pages. To appear in Classical and
Quantum Gravity. (uses iopart.cls
Letter to the editor. Novel finite-differencing techniques for numerical relativity: application to black hole excision
We use rigorous techniques from numerical analysis of hyperbolic equations in bounded domains to construct stable finite-difference schemes for Numerical Relativity, in particular for their use in black hole excision. As an application, we present 3D simulations of a scalar field propagating in a Schwarzschild black hole background
Evolutions in 3D numerical relativity using fixed mesh refinement
We present results of 3D numerical simulations using a finite difference code featuring fixed mesh refinement (FMR), in which a subset of the computational domain is refined in space and time. We apply this code to a series of test cases including a robust stability test, a nonlinear gauge wave and an excised Schwarzschild black hole in an evolving gauge. We find that the mesh refinement results are comparable in accuracy, stability and convergence to unigrid simulations with the same effective resolution. At the same time, the use of FMR reduces the computational resources needed to obtain a given accuracy. Particular care must be taken at the interfaces between coarse and fine grids to avoid a loss of convergence at higher resolutions, and we introduce the use of 'buffer zones' as one resolution of this issue. We also introduce a new method for initial data generation, which enables higher order interpolation in time even from the initial time slice. This FMR system, 'Carpet', is a driver module in the freely available Cactus computational infrastructure, and is able to endow generic existing Cactus simulation modules ('thorns') with FMR with little or no extra effort