ABSTRACT Hydromagnetic turbulence produced during phase transitions in the early universe can be a powerful source of stochastic gravitational waves (GWs). GWs can be modelled by the linearised spatial part of the Einstein equations sourced by the Reynolds and Maxwell stresses. We have implemented two different GW solvers into the Pencil Code – a code which uses a third order timestep and sixth order finite differences. Using direct numerical integration of the GW equations, we study the appearance of a numerical degradation of the GW amplitude at the highest wavenumbers, which depends on the length of the timestep – even when the Courant–Friedrichs–Lewy condition is ten times below the stability limit. This degradation leads to a numerical error, which is found to scale with the third power of the timestep. A similar degradation is not seen in the magnetic and velocity fields. To mitigate numerical degradation effects, we alternatively use the exact solution of the GW equations under the assumption that the source is constant between subsequent timesteps. This allows us to use a much longer timestep, which cuts the computational cost by a factor of about ten.

中文翻译：

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求解水磁湍流引力波方程的时间步长约束

摘要 早期宇宙相变期间产生的水磁湍流可能是随机引力波 (GW) 的强大来源。GW 可以通过由雷诺和麦克斯韦应力产生的爱因斯坦方程的线性空间部分建模。我们在 Pencil Code 中实现了两种不同的 GW 求解器——一种使用三阶时间步长和六阶有限差分的代码。使用 GW 方程的直接数值积分，我们研究了最高波数下 GW 振幅数值退化的出现，这取决于时间步长的长度——即使 Courant-Friedrichs-Lewy 条件低于稳定性十倍限制。这种退化会导致数值误差，发现它与时间步长的三次方成比例。在磁场和速度场中没有看到类似的退化。为了减轻数值退化效应，我们或者在假设源在后续时间步长之间保持恒定的情况下使用 GW 方程的精确解。这使我们能够使用更长的时间步长，从而将计算成本降低约 10 倍。