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.

翻译：紧致有限差分格式通过隐式方式定义导数近似，为实现区域分解并行性需要对线性系统进行适当分区。对于波动传播问题，保持精度阶、色散和耗散的一致性至关重要，以避免离散问题关联谱的严重畸变。本研究通过固定形式精度阶进行数值谱误差调谐，自主设计了新型紧致有限差分格式。网格收敛性测试表明，相比标准有限差分方法，误差可降低至少一个量级。提出的混合匹配-通信策略可在区域分解下维持上述特性。基于指数积分或显式龙格-库塔方法的线性波动传播问题演化表明，改进效果保持稳健。首次证明紧致有限差分可应用于数值相对论的Z4c公式，我们将仅含头文件的模板化C++实现与高性能GR-Athena++代码耦合。在测试基准问题上演化Z4c时，相比标准有限差分，传播度量分量的相位误差降低至少一个量级。同时展示了采用紧致有限差分的稳定双黑洞演化及其改进的收敛性。