带区域分解的谱调谐紧致差分格式及其在数值相对论中的应用
Spectrally-tuned compact finite-difference schemes with domain decomposition and applications to numerical relativity
摘要：
紧凑的有限差分（FD）格式隐含地指定了导数近似，因此为了实现与区域分解的并行性，需要对线性系统进行适当的划分。一致的精度，色散和耗散的顺序是至关重要的，以保持在波传播问题，这样的离散化问题的相关频谱的变形是不太严重。在这项工作中，我们考虑数值调谐谱误差，在固定的正式顺序的准确性，自动设计新的紧凑FD计划。网格收敛测试表明，误差减少至少一个数量级以上的标准FD。一个建议的混合匹配通信策略保持上述属性下域分解。根据线性波传播问题的发展，利用指数积分或显式龙格-库塔方法的改进被发现保持稳健。第一个演示，紧凑FD方法可以应用到Z4 c制定的数值相对论，我们耦合我们的头只，模板化的C++实现的高性能GR-雅典娜++代码。在试验台问题上的Z4 c的发展表明，与传播度量分量的FD相比，相位误差至少减少了一个数量级。稳定的二进制黑洞的演化，利用紧凑的FD与改进的收敛性也被证明。
英文摘要：
Compact finite-difference (FD) schemes specify derivative approximations implicitly, thus to achieve parallelism with domain-decomposition suitable partitioning of linear systems is required. Consistent order of accuracy, dispersion, and dissipation is crucial to maintain in wave propagation problems such that deformation of the associated spectra of the discretized problems is not too severe. In this work we consider numerically tuning spectral error, at fixed formal order of accuracy to automatically devise new compact FD schemes. Grid convergence tests indicate error reduction of at least an order of magnitude over standard FD. A proposed hybrid matching-communication strategy maintains the aforementioned properties under domain-decomposition. Under evolution of linear wave-propagation problems utilizing exponential integration or explicit Runge-Kutta methods improvement is found to remain robust. A first demonstration that compact FD methods may be applied to the Z4c formulation of numerical relativity is provided where we couple our header-only, templated C++ implementation to the highly performant GR-Athena++ code. Evolving Z4c on test-bed problems shows at least an order in magnitude reduction in phase error compared to FD for propagated metric components. Stable binary-black-hole evolution utilizing compact FD together with improved convergence is also demonstrated.
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