| Abstract: |
| A high-frequency recovered fully discrete low-regularity integrator is constructed to approximate rough and possibly discontinuous solutions of the semilinear wave equation. The proposed method, with high-frequency recovery techniques, can capture the discontinuities of the solutions correctly without spurious oscillations and 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 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 stepsize condition $\tau =O(1/N)$, where $\tau$ and $N$ denote the time stepsize and the number of Fourier terms in the space discretization, respectively. 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. |
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