  A fully discrete low-regularity integrator with high-frequency recovery
techniques is constructed to approximate rough and possibly discontinuous
solutions of the semilinear wave equation. The proposed method can capture the
discontinuities of the solutions correctly without spurious oscillations and
can approximate rough and discontinuous solutions with a higher convergence
rate than pre-existing methods. Rigorous analysis is presented for the
convergence rates of the proposed method in approximating solutions such that
$(u,\partial_{t}u)\in C([0,T];H^{\gamma}\times H^{\gamma-1})$ for
$\gamma\in(0,1]$. For discontinuous solutions of bounded variation in one
dimension (which allow jump discontinuities), the proposed method is proved to
have almost first-order convergence under the step size condition $\tau \sim
N^{-1}$, where $\tau$ and $N$ denote the time step size and the number of
Fourier terms in the space discretization, respectively. Extensive numerical
examples are presented in both one and two dimensions to illustrate the
advantages of the proposed method in improving the accuracy in approximating
rough and discontinuous solutions of the semilinear wave equation. The
numerical results are consistent with the theoretical results and show the
efficiency of the proposed method.
