Summation by parts and truncation error matching on hyperboloidal slices

Shalabh Gautam, Alex Van'o-Vinuales, D. Hilditch, S. Bose
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引用次数: 6

Abstract

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.
双曲面切片的部分求和与截断误差匹配
我们研究了部分求和(SBP)数值格式的稳定性,这些格式使用双曲面切片在计算域中包含未来的零无穷。这种包含有助于减轻外边界效应,并在未来,将有助于减少重力波形提取的系统误差。我们还研究了一个截断误差匹配的设置。我们的SBP-Stable格式保证了一类线性波动方程在半离散水平上的能量平衡。我们还开发了专门的耗散算子。整个构造是在球对称的二阶精度下进行的,但可以直接推广到不对称的高阶或谱精度。在实际实现中,我们首先演化出一个服从线性波动方程的标量场,并观察到预期的长期稳定性和范数收敛性。对于势项,我们得到类似的结果。为了检验该方法的局限性,我们考虑了一个大质量场,其运动方程不正则化,其动力学接近零无穷大,其中涉及无法由代码解析的激发传入脉冲,与无质量设置非常不同。我们仍然观察到良好的能量守恒,但收敛性并不令人满意。总的来说,我们的结果表明,当渐近解空间接近波动方程的解空间时,紧化双曲面切片可能是可证明有效的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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