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 analogous convergence results, where the order is reduced by 1/2 in both cases. 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.
