  We examine stability of summation by parts (SBP) numerical schemes that use
hyperboloidal slices to include future null infinity in the computational
domain. This inclusion serves to mitigate outer boundary effects and, in the
future, will help reduce systematic errors in gravitational waveform
extraction. We also study a setup with truncation error matching. Our
SBP-Stable scheme guarantees energy-balance for a class of linear wave
equations at the semidiscrete level. We develop also specialized dissipation
operators. The whole construction is made at second order accuracy in spherical
symmetry, but could be straightforwardly generalized to higher order or
spectral accuracy without symmetry. In a practical implementation we evolve
first a scalar field obeying the linear wave equation and observe, as expected,
long term stability and norm convergence. We obtain similar results with a
potential term. To examine the limitations of the approach we consider a
massive field, whose equations of motion do not regularize, and whose dynamics
near null infinity, which involve excited incoming pulses that can not be
resolved by the code, is very different to that in the massless setting. We
still observe excellent energy conservation, but convergence is not
satisfactory. Overall our results suggest that compactified hyperboloidal
slices are likely to be provably effective whenever the asymptotic solution
space is close to that of the wave equation.
