Discontinuous Galerkin method for blow-up solutions of nonlinear 1D wave equations

TL;DR

This paper develops a discontinuous Galerkin method for 1D nonlinear wave equations, demonstrating stability on nonuniform meshes and accurately approximating blow-up times through theoretical analysis and numerical validation.

Contribution

It introduces a stable discontinuous Galerkin scheme for nonlinear wave equations that effectively captures blow-up phenomena and converges to the theoretical blow-up time.

Findings

The scheme is stable with nonuniform time meshes.

Numerical blow-up times approximate theoretical values.

Numerical examples confirm theoretical results.

Abstract

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.