We study a variant of the Strang splitting for the time integration of the semilinear wave equation under the finite-energy condition on the torus $\mathbb{T}^3$. In the case of a cubic nonlinearity, we show almost second-order convergence in $L^2$ and almost first-order convergence in $H^1$. If the nonlinearity has a quartic form instead, we show an analogous convergence result with an order reduced by 1/2. To our knowledge these are the best convergence results available for the 3D cubic and quartic wave equations under the finite-energy condition. Our approach relies on continuous- and discrete-time Strichartz estimates. We also make use of the integration and summation by parts formulas to exploit cancellations in the error terms. Moreover, error bounds for a full discretization using the Fourier pseudo-spectral method in space are given. Finally, we discuss a numerical example indicating the sharpness of our theoretical results.

翻译：我们研究了环面$\mathbb{T}^3$上有限能量条件下半线性波动方程时间积分的Strang分裂法的一种变体。对于三次非线性情形，我们证明了$L^2$范数下的几乎二阶收敛性以及$H^1$范数下的几乎一阶收敛性。若非线性项为四次形式，我们给出了类似的收敛结果，但收敛阶降低1/2。据我们所知，这是在有限能量条件下三维三次和四次波动方程目前可获得的最佳收敛结果。我们的方法依赖于连续时间和离散时间的Strichartz估计，并利用分部积分与求和公式来挖掘误差项中的相消效应。此外，我们给出了空间上采用傅里叶伪谱方法进行全离散化的误差界。最后，我们通过数值算例讨论了理论结果的尖锐性。