The Einstein field equations of gravitation are characterized by cross-scale, high-order nonlinear terms, representing a challenge for numerical modeling. In an exact spectral decomposition, high-order nonlinearities correspond to a convolution that numerically might lead to aliasing instabilities. We present a study of this problem, in vacuum conditions, based on the \(3+1\) Baumgarte–Shibata–Shapiro–Nakamura (BSSN) formalism. We inspect the emergence of numerical artifacts, in a variety of conditions, by using the Spectral-FIltered Numerical Gravity codE (SFINGE)—a pseudo-spectral algorithm, based on a classical (Cartesian) Fourier decomposition. By monitoring the highest \(k-\)modes of the dynamical fields, we identify the culprits of the aliasing and propose procedures that cure such instabilities, based on the suppression of the aliased wavelengths. This simple algorithm, together with appropriate treatment of the boundary conditions, can be applied to a variety of gravitational problems, including those related to massive objects dynamics.

爱因斯坦引力场方程的特点是跨尺度、高阶非线性项，这对数值建模提出了挑战。在精确的频谱分解中，高阶非线性对应于在数值上可能导致混叠不稳定性的卷积。我们在真空条件下基于\(3+1\) Baumgarte-Shibata-Shapiro-Nakamura (BSSN) 形式主义对这个问题进行了研究。我们通过使用光谱滤波数值重力编码 ( SFINGE )——一种基于经典（笛卡尔）傅立叶分解的伪光谱算法，检查各种条件下数值伪影的出现。通过监测最高\(k-\)根据动态场的模式，我们确定了混叠的罪魁祸首，并基于混叠波长的抑制提出了解决这种不稳定性的程序。这种简单的算法，加上对边界条件的适当处理，可以应用于各种重力问题，包括与大质量物体动力学相关的问题。