Long term black hole evolution with the BSSN system by pseudospectral methods (0911.0973v2)

Abstract: We present long term evolutions of a single black hole of mass $M$ with the BSSN system using pseudospectral methods. For our simulations we use the SGRID code where the BSSN system is implemented in its standard second order in space form. Previously we found that such simulations are quite unstable. The main goal of this paper is to present two improvements which now allow us to evolve for longer times. The first improvement is related to the boundary conditions at the excised black hole interior. We now use a gauge condition that ensures that all modes are going into the black hole, so that no boundary conditions are needed at the excision surface. The second more significant improvement has to do with our particular numerical method and involves filters based on projecting the double Fourier expansions used for the angular dependence onto Spherical Harmonics. With these two improvements it is now easily possible to evolve for several thousand $M$. The only remaining limitation seems to be the radiative outer boundary conditions used here. Yet this problem can be ameliorated by pushing out the location of the outer boundary, which leads to even longer run-times.

• The paper introduces innovative gauge boundary conditions that eliminate the need for excision boundaries, enhancing simulation stability.
• The paper employs spectral filtering via spherical harmonic projections to suppress non-physical modes, enabling evolutions to exceed 3244M.
• The paper highlights the potential of the BSSN formulation with pseudospectral methods while underscoring the need for refined outer boundary conditions.

Long-Term Black Hole Evolution Using the BSSN System and Pseudospectral Methods

The paper conducted by Wolfgang Tichy focuses on the long-term evolution of single black holes using the BSSN (Baumgarte–Shapiro–Shibata–Nakamura) formulation of the Einstein equations implemented via pseudospectral methods. This approach contrasts with the more widely utilized finite differencing techniques in numerical relativity. The research aims to investigate the feasibility of using spectral methods for evolving black holes over extended periods, introducing improvements that significantly enhance stability and runtime.

Methodological Advancements

The paper outlines two major improvements that facilitate longer simulation times:

1. Boundary Condition Innovation: The author introduces a gauge condition that ensures inward propagation of all modes at the excision boundary, thus eliminating the need for boundary conditions at this surface. This adjustment is critical for maintaining stability in the evolution of black holes using spectral methods.
2. Spectral Filtering Techniques: It is detailed how the application of filters based on spherical harmonic projections from double Fourier expansions can remove non-physical modes and reduce high-frequency noise. This filtering corrects the instability issues previously encountered, enabling the system to evolve for several thousand mass units (M) before numerical errors become significant.

Simulation Outcomes

The results demonstrate a marked improvement over prior attempts with pseudospectral methods:

• Extended Run-Times: The simulations now sustain for over 3244M at a high radial resolution, a stark contrast to the previous threshold of around 100M. The improvement evidences the effectiveness of the newly implemented methods in maintaining stability over long periods.
• Resolution and Boundary Effects: The simulations indicate that runtime increases with higher radial resolution. The challenge remains at the outer boundary, where radiative boundary conditions introduce instabilities. Increasing the radius of the outer boundary offers a practical workaround, extending runtimes even further, albeit at the cost of computational resources.

Challenges and Implications

The paper highlights the persistent issues associated with simplistic outer boundary conditions, such as radiative conditions that inadequately address the inflow/outflow of modes. These constraints lead to the introduction of errors from the domain boundaries, manifesting over extended periods as the simulations progress.

The successes detailed in Tichy's work suggest the potential for pseudospectral methods to perform effective long-term evolutions without reliance on finite differencing. However, it also calls attention to the necessity for better theoretical formulations of boundary conditions within the BSSN framework, a task that remains open for further exploration in the numerical relativity community.

Future Directions

The research underscores several avenues for future investigation:

• Development of Robust Boundary Conditions: Continued efforts are needed to derive conditions that ensure well-posedness and stability, potentially looking to advances in harmonic coordinates and alternative formulations of relativistic field equations.
• Applications to Binary Systems: With improved stability, extending these methods to the evolution of binary black hole systems under the BSSN formulation via spectral methods presents a promising frontier.
• Harmony Between Computational Techniques: Exploring hybrid models that leverage the strengths of different numerical methodologies may offer pathways to optimizing both stability and efficiency in evolving complex spacetime geometries.

Tichy's work provides a pivotal step towards enhancing the toolset available for numerical simulations in general relativity, offering insights that can inform both theoretical and computational advancements in the field.