Unconditional stability and optimal error estimates of discontinuous Galerkin methods for the second-order wave equation

Abstract

In this paper, we revisit the numerical solution of the scalar second-order wave equation by discontinuous Galerkin methods. The numerical methods are different from the ones found in existing literature. Moreover, we provide a stability analysis and derive optimal order error estimates through a more direct approach. The error estimate in an $H^1\left(\mathrm{\Omega }\right)$-like norm is derived based on an analysis of the truncation error while that in the $L^2\left(\mathrm{\Omega }\right)$ norm based on an application of the Aubin-Nitsche technique. Numerical simulation results are reported in support of the theoretical error estimates.

Keywords:

• Second-order wave equation
• discontinuous Galerkin methods
• fully discrete approximation
• Gronwall inequality

AMS Classifications:

• 65N30
• 49J40

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by the National Natural Science Foundation of China [grant numbers 61663035,11771350], Inner Mongolia Natural Science Foundation of China [grant number 2018MS06017], and NSF [grant number DMS-1521684].