We develop and study a time-space discrete discontinuous Galerkin finite elements method to approximate the solution of one-dimensional nonlinear wave equations. We show that the numerical scheme is stable if a nonuniform time mesh is considered. We also investigate the blow-up phenomena and we prove that under weak convergence assumptions, the numerical blow-up time tends toward the theoretical one. The validity of our results is confirmed throughout several numerical examples and benchmarks.

翻译：我们开发并研究一种时间空间离散不连续的 Galerkin 有限元素方法, 以近似单维非线性波等式的解决方案。 我们显示, 如果考虑的是非统一时间网格, 数字方案是稳定的。 我们还调查了爆炸现象, 我们证明, 在衰弱的趋同假设下, 数字打击时间倾向于理论。 我们的结果的有效性通过几个数字例子和基准得到了确认 。