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.
Free Hyperboloidal Evolution of the Einstein-Maxwell-Klein-Gordon System
CENTRA, Departamento de Física do Instituto Superior Técnico (IST), Universidade de Lisboa（里斯本大学高等理工学院物理系CENTRA）；Instituto de Telecomunicações (IT), Universidade de Aveiro（阿威罗大学电信研究所）；Departament de Física, Universitat de les Illes Balears, IAC3（巴利阿里群岛大学物理系IAC3）
3d Summation-by-Parts scheme for Linear Wave Equations on Hyperboloidal Slices
Indian Institute of Science Education and Research (IISER) Pune（印度科学教育研究所浦那分校）；International Centre for Theoretical Sciences (ICTS), Tata Institute of Fundamental Research (TIFR)（塔塔基础研究所国际理论科学中心）；Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京数学科学与应用研究院）
