Numerical relativity convergence and error budgets

Suppose a dimensionless observable has a smooth asymptotic error expansion with leading coefficient C times spacing to power p. A three-grid study with a common refinement ratio can estimate p from successive differences, provided the leading term dominates. The physical initial data, evaluation event, domain, gauge prescription and extraction procedure must be matched. Changing those choices between runs can turn a purported convergence ratio into a comparison of different problems.

For waveforms, alignment can remove conventional time and phase offsets, but its interval and objective must be stated. Excessive alignment freedom can hide a meaningful accumulated phase error. Spatial, temporal and elliptic-solver errors should be refined or controlled consistently; otherwise the observed order may saturate at the largest unchanged error source. A formal high-order stencil does not guarantee the same order for an extracted quantity after interpolation, boundaries and integration.

With a justified leading order, Richardson extrapolation cancels its leading term. The difference between the finest value and extrapolate estimates the remaining leading discretization error, rather than proving a strict bound. Additional resolutions and changed fitting windows test the expansion. Irregular convergence, especially near regridding or merger, may require a more conservative assessment than quoting one fitted power with many decimal places.

Constraints and global balances provide independent consistency information. In an asymptotically flat vacuum spacetime with the final remnant at rest in the comparison frame, initial ADM energy can be compared with remnant mass plus radiated energy at infinity under compatible definitions. A recoiling remnant instead requires its total energy, including kinetic energy. Finite boundaries, finite-radius fluxes, residual radiation on the grid and initial spurious radiation complicate that comparison. Horizon spin estimates likewise need their own symmetry and surface-finding controls. Good balance is valuable evidence, but correlated numerical errors can sometimes cancel in a balance residual.

A practical budget may track grid discretization, initial-data tolerance, finite outer boundary, wave extraction, waveform integration and horizon diagnostics. Vary each controlling choice enough to identify its influence on the target observable. These contributions can be correlated; treating them all as independent random standard deviations is usually unjustified. Deterministic absolute bounds can safely be added, while a root-sum-square rule needs a defensible statistical model or covariance information.

Record the parameter range, duration and observable for which the assessment applies. A phase error at one frequency does not automatically bound all harmonics or a derived detector mismatch. A result that passes every available check still has assumptions, but explicitly measured sensitivities make those assumptions reviewable. The synthetic examples below practice the arithmetic of extrapolation and a worst-case balance budget; they do not certify any actual binary simulation or establish continuum existence from finite data.