A local energy-based discontinuous Galerkin method for fourth order semilinear wave equations

TL;DR

This paper introduces a novel energy-based discontinuous Galerkin method tailored for fourth-order semilinear wave equations, offering improved efficiency, stability, and optimal convergence, with practical numerical validation.

Contribution

It extends the energy-based DG approach to higher-order wave equations, achieving unconditionally stable, computationally efficient schemes with proven convergence.

Findings

More computationally efficient than local DG methods

Unconditionally stable without penalty terms

Achieves optimal convergence in $L^2$ norm

Abstract

This paper generalizes the earlier work on the energy-based discontinuous Galerkin method for second-order wave equations to fourth-order semilinear wave equations. We first rewrite the problem into a system with a second-order spatial derivative, then apply the energy-based discontinuous Galerkin method to the system. The proposed scheme, on the one hand, is more computationally efficient compared with the local discontinuous Galerkin method because of fewer auxiliary variables. On the other hand, it is unconditionally stable without adding any penalty terms, and admits optimal convergence in the $L^2$ norm for both solution and auxiliary variables. In addition, the energy-dissipating or energy-conserving property of the scheme follows from simple, mesh-independent choices of the interelement fluxes. We also present a stability and convergence analysis along with numerical experiments to demonstrate optimal convergence for certain choices of the interelement fluxes.