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 c formulation of numerical relativity is provided where we couple our header-only, templated implementation to the highly performant code. Evolving c 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.

中文翻译：

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具有域分解的光谱调谐紧凑有限差分格式及其在数值相对论中的应用

紧凑有限差分（FD）方案隐式指定导数近似，因此为了实现域分解的并行性，需要对线性系统进行适当的划分。精度、色散和耗散的一致顺序对于维持波传播问题至关重要，这样离散问题的相关谱的变形就不会太严重。在这项工作中，我们考虑以固定的精度形式数值调整光谱误差，以自动设计新的紧凑型 FD 方案。网格收敛测试表明，与标准 FD 相比，误差至少减少了一个数量级。所提出的混合匹配通信策略在域分解下保持上述属性。在线性波传播问题的演化过程中，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。我们将纯头文件、模板化实现与高性能代码结合起来，首次演示了紧凑的 FD 方法可以应用于数值相对论的 c 公式。测试台问题上的演进表明，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。还展示了利用紧凑 FD 以及改进的收敛性的稳定双黑洞演化。