In this work, we develop and study a discontinuous Galerkin (DG) method to approximate the solution of 1D nonlinear wave equations. We show that the numerical scheme is consistent, stable and convergent if a nonuniform time mesh is considered. We also investigate the blow-up phenomena and we prove that  the numerical blow-up time converges toward the theoretical one. The validity of our results is confirmed throughout several numerical examples and benchmarks.
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.
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In this work, we
We
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discontinuous Galerkin (DG)
time-space discrete discontinuous Galerkin finite elements
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1D
one-dimensional
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consistent, stable and convergent
stable
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under weak convergence assumptions,
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converges
tends