Linearly Implicit and High-Order Energy-Conserving Schemes for Nonlinear Wave Equations

A key issue in developing efficient numerical schemes for nonlinear wave equations is the energy-conserving. Most existing schemes of the energy-conserving are fully implicit and the schemes require an extra iteration at each time step and considerable computational cost for a long time simulation, while the widely-used q-stage (implicit) Gauss scheme (method) only preserves polynomial Hamiltonians up to degree 2q. In this paper, we present a family of linearly implicit and high-order energy-conserving schemes for solving nonlinear wave equations. The construction of schemes is based on recently-developed scalar auxiliary variable technique with a combination of classical high-order Gauss methods and extrapolation approximation. We prove that the proposed schemes are unconditionally energy-conserved for a general nonlinear wave equation. Numerical results are given to show the energy-conserving and the effectiveness of schemes.