An arXiv preprint dated 26 Aug 2026 reports a fully 3D numerical scheme that evolved a class of linear wave equations on hyperboloidal slices of a fixed Minkowski background. The calculation included grid points at the origin, the z-axis and infinity, and the reported tests showed stable evolution, non-increasing discrete energy and convergence that approached second order.
Putting infinity on the grid
The proposed Summation-by-Parts, or SBP, scheme uses compactification and rescaling to include infinity in the calculation despite coordinate singularities. It also uses generalized Kreiss-Oliger dissipation operators, numerical filters defined throughout the domain, including boundary points, with a dissipative energy-norm property. In the implementation, time was advanced with standard Runge-Kutta integration known as RK4, and space was represented with second-order finite differences. Higher-order finite-difference and pseudo-spectral methods are described as possible alternatives.
A choice at the outer boundary
At the outer boundary, the edge representing infinity, the study compared two variants. SBP-TEM preserves numerical accuracy across the domain, whereas SBP-Stable guarantees stability and negative-definiteness of the energy flux at I+, the notation for future null infinity. That creates a trade-off between retaining accuracy and enforcing the stronger energy-flux guarantee.
Three tests, with mixed demands
The numerical study tested F = 0, F = 1/χ2 and F = m2 with both discretizations. All simulations reportedly maintained stable evolution across the considered resolutions up to CFL = 2.6785, the empirical maximum reported, and remained stable without dissipation at a = 0 despite numerical noise. To reduce high-frequency numerical noise, the paper reports minimal artificial-dissipation coefficients of a = 0.002 for SBP-TEM and a = 0.008 for SBP-Stable.
What the wave tests showed
In the F = 0 case, a narrow pulse split into incoming and outgoing components. The outgoing component moved at unit speed, and the signal reached I+ in two distinct bursts. Both discrete energy norms were non-increasing and closely aligned with the continuum energy from a closed-form solution.
Accuracy under refinement
At higher base resolutions, the energy-norm convergence order in F = 0 approached 2 and remained close to 2 along the three coordinate directions at long times. The E1 and E2 norms used information from all grid points across all resolutions, and they did not show the loss of convergence-order correlation seen in conventional coarse-grid comparisons.
Tails and a singular case
For F = 1/χ2, the late-time signal at I+ showed power-law decay. Fitted slopes moved toward a constant as resolution increased, but the paper does not establish that limiting slope as a final measured value.
In the F = m2 case, the solution remained inside the light cone and evolved stably. With dissipation turned off, total discrete energy was conserved at all times despite singular potential terms at I+. Angular convergence was perfect second order, but radial convergence was compromised by the singular behavior at infinity.
What remains untested
The evidence is confined to a class of linear wave equations on a fixed Minkowski background. It does not establish how the scheme would perform for fully nonlinear Einstein equations. The implementation uses second-order finite differences and RK4; higher-order finite-difference and pseudo-spectral methods are possible alternatives, but their performance is not tested here.
Paper data and sources
Original title: A 3D Summation-by-Parts scheme on a Hyperboloidal Foliation of Minkowski
Authors: Shalabh Gautam
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text