A pseudo-spectral numerical approach to solve the Einstein field equations

In this paper, we present a numerical study of the Einstein field equations, based on the 3 + 1 foliation of the spacetime. A pseudo-spectral technique has been employed for simulations in vacuum conditions, within the formalism of Baumgarte-Shapiro-Shibata-Nakamura (BSSN). We use the Spectral-FIltered Numerical Gravity codE (SFINGE), a numerical code based on the Fourier decomposition, accompanied by different filtering techniques. The accuracy of the model has been validated through standard testbeds, revealing that the filtered pseudo-spectral technique is incredibly accurate. We evolved black hole dynamics in vacuum conditions, in small domains, making use of hyperviscous dissipation that suppresses spurious boundary problems. This simple algorithm can be applied to a variety of gravitational problems, including those related to massive objects dynamics.