Numerical experiments of adjusted Baumgarte-Shapiro-Shibata-Nakamura systems for controlling constraint violations

We present our numerical comparisons between the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation widely used in numerical relativity today and its adjusted versions using constraints. We performed three test beds: gauge-wave, linear wave, and Gowdy-wave tests, proposed by the Mexico workshop on the formulation problem of the Einstein equations. We tried three kinds of adjustments, which were previously proposed from the analysis of the constraint propagation equations, and investigated how they improve the accuracy and stability of evolutions. We observed that the signature of the proposed Lagrange multipliers are always right and the adjustments improve the convergence and stability of the simulations. When the original BSSN system already shows satisfactory good evolutions (e.g., linear wave test), the adjusted versions also coincide with those evolutions, while in some cases (e.g., gauge-wave or Gowdy-wave tests) the simulations using the adjusted systems last 10 times as long as those using the original BSSN equations. Our demonstrations imply a potential to construct a robust evolution system against constraint violations even in highly dynamical situations.