On Two Conservative HDG Schemes for Nonlinear Klein-Gordon Equation

TL;DR

This paper introduces and analyzes two conservative HDG schemes for the nonlinear Klein-Gordon equation, providing error estimates, convergence orders, and energy conservation proofs, supported by numerical experiments.

Contribution

It proposes and rigorously analyzes two new conservative HDG schemes for the nonlinear Klein-Gordon equation, including error estimates and energy conservation properties.

Findings

Flux and displacement approximations converge with order O(h^{k+1})

Post-processed solutions achieve order O(h^{k+2}) convergence

Discrete energy is conserved with optimal error estimates

Abstract

In this article, a hybridizable discontinuous Galerkin (HDG) method is proposed and analyzed for the Klein-Gordon equation with local Lipschitz-type non-linearity. {\it A priori} error estimates are derived, and it is proved that approximations of the flux and the displacement converge with order $O(h^{k+1}),$ where $h$ is the discretizing parameter and $k$ is the degree of the piecewise polynomials to approximate both flux and displacement variables. After post-processing of the semi-discrete solution, it is shown that the post-processed solution converges with order $O(h^{k+2})$ for $k \geq 1.$ Moreover, a second-order conservative finite difference scheme is applied to discretize in time %second-order convergence in time. and it is proved that the discrete energy is conserved with optimal error estimates for the completely discrete method. %Since at each time step, one has to solve a nonlinear system of algebraic equations, To avoid solving a nonlinear system of algebraic equations at each time step, a non-conservative scheme is proposed, and its error analysis is also briefly established. Moreover, another variant of the HDG scheme is analyzed, and error estimates are established. Finally, some numerical experiments are conducted to confirm our theoretical findings.