Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stability analysis and improvement of the covariant BSSN formulation against the FLRW spacetime background

Hidetomo Hoshino, Takuya Tsuchiya, Gen Yoneda

TL;DR

The paper analyzes the numerical stability of the covariant BSSN (cBSSN) formulation on FLRW spacetimes by examining the constraint propagation equations and their constraint amplification factors (CAFs). It shows that the unadjusted system exhibits unstable CAFs (three positive roots) and demonstrates that adding a damped term to the evolution of the $ar{oldsymbol{ abla}}^i$ variable with a negative damping parameter $\kappa$ reduces CAFs, improving stability. The authors validate the analytical predictions with numerical simulations in matter-, vacuum-, and radiation-dominated universes, finding the negative-damping case minimizes constraint violations. This work provides a principled method to enhance long-term cosmological simulations in numerical relativity, enabling more reliable modeling of near-FLRW spacetimes and perturbations therein.

Abstract

In this study, we investigate the numerical stability of the covariant Baumgarte--Shapiro--Shibata--Nakamura (cBSSN) formulation against the Friedmann--Lemaître--Robertson--Walker spacetime. To evaluate the numerical stability, we calculate the constraint amplification factor by the eigenvalue analysis of the evolution of the constraint. We propose a modification to the time evolution equations of the cBSSN formulation for a higher numerical stability. Furthermore, we perform numerical simulations using the modified formulation to confirm its improved stability.

Stability analysis and improvement of the covariant BSSN formulation against the FLRW spacetime background

TL;DR

The paper analyzes the numerical stability of the covariant BSSN (cBSSN) formulation on FLRW spacetimes by examining the constraint propagation equations and their constraint amplification factors (CAFs). It shows that the unadjusted system exhibits unstable CAFs (three positive roots) and demonstrates that adding a damped term to the evolution of the variable with a negative damping parameter reduces CAFs, improving stability. The authors validate the analytical predictions with numerical simulations in matter-, vacuum-, and radiation-dominated universes, finding the negative-damping case minimizes constraint violations. This work provides a principled method to enhance long-term cosmological simulations in numerical relativity, enabling more reliable modeling of near-FLRW spacetimes and perturbations therein.

Abstract

In this study, we investigate the numerical stability of the covariant Baumgarte--Shapiro--Shibata--Nakamura (cBSSN) formulation against the Friedmann--Lemaître--Robertson--Walker spacetime. To evaluate the numerical stability, we calculate the constraint amplification factor by the eigenvalue analysis of the evolution of the constraint. We propose a modification to the time evolution equations of the cBSSN formulation for a higher numerical stability. Furthermore, we perform numerical simulations using the modified formulation to confirm its improved stability.
Paper Structure (9 sections, 46 equations, 3 figures)

This paper contains 9 sections, 46 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Review of cBSSN formulation
  3. Review of CAF and adjusted system
  4. CAFs against the FLRW spacetime
  5. Adjusted system for the FLRW spacetime
  6. Numerical results
  7. Summary
  8. Numerical results: vacuum-dominated universe
  9. Numerical results: radiation-dominated universe

Figures (3)

  • Figure 1: The horizontal axis represents time and the vertical axis represents the value of $\log_{10}{C}$. The value with $\kappa=-0.1$ is the smallest of the three.
  • Figure 2: The horizontal axis represents time and the vertical axis represents the constraint error. The error with $\kappa=-0.1$ is the smallest of the three.
  • Figure 3: The horizontal axis represents time and the vertical axis represents the constraint error. The error with $\kappa=-0.1$ is the smallest of the three.