Unconditionally optimal convergence of an energy-conserving and linearly implicit scheme for nonlinear wave equations

TL;DR

This paper introduces an energy-conserving, linearly implicit scheme for nonlinear wave equations, achieving optimal convergence without restrictions on time-step size, and confirms its effectiveness through numerical experiments.

Contribution

It provides a novel analysis directly estimating solution bounds in $H^2$-norm, leading to unconditional optimal error estimates for the scheme.

Findings

Energy-conserving property confirmed by numerical tests

Unconditional convergence demonstrated

Optimal error estimates validated numerically

Abstract

In this paper, we present and analyze an energy-conserving and linearly implicit scheme for solving the nonlinear wave equations. Optimal error estimates in time and superconvergent error estimates in space are established without time-step dependent on the spatial mesh size. The key is to estimate directly the solution bounds in the $H^2$-norm for both the nonlinear wave equation and the corresponding fully discrete scheme, while the previous investigations rely on the temporal-spatial error splitting approach. Numerical examples are presented to confirm energy-conserving properties, unconditional convergence, and optimal error estimates, respectively, of the proposed fully discrete schemes.