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.

翻译：通过构造结合高频恢复技术的全离散低正则性积分器，对半线性波动方程的粗糙及可能间断解进行逼近。所提方法能正确捕捉解的不连续性而不产生虚假振荡，且相较已有方法能以更高收敛率逼近粗糙与间断解。针对满足$(u,\partial_{t}u)\in C([0,T];H^{\gamma}\times H^{\gamma-1})$（$\gamma\in(0,1]$）的解，给出了所提方法收敛率的严格分析。对一维有界变差（允许跳跃间断）的间断解，证明了在时间步长条件$\tau \sim N^{-1}$（其中$\tau$为时间步长，$N$为空间离散傅里叶项数）下，该方法具有近似一阶收敛性。通过一维与二维的大量数值算例，展示了所提方法在提高半线性波动方程粗糙与间断逼近精度方面的优势。数值结果与理论分析一致，验证了方法的有效性。