Fully decoupled and high-order linearly implicit energy-preserving RK-GSAV methods for the coupled nonlinear wave equation

Abstract

This paper is concerned with high-order accurate, linearly implicit and energy-preserving schemes for the coupled nonlinear wave equation. To this end, a novel auxiliary variable approach proposed in a recent paper (Ju et al., 2022) is applied to reformulate the governing equation in an equivalent system, which possesses a modified energy that consists of primary functional and quadratic functional. Subsequently, we utilize the extrapolation strategy/prediction–correction technique for treating the nonlinear terms of the equivalent system to obtain a linearized energy-preserving system. Then, a series of fully decoupled high-order linear energy-preserving schemes are constructed by using the symplectic Runge–Kutta (RK) methods. We show that the modified energy functional can be precisely conserved under certain circumstances for the coefficients of the symplectic RK methods, and the unique solvability of the proposed time-stepping scheme is analyzed. In numerical implementations, a class of linear algebraic equations with constant-coefficient matrices need to be solved at each time step and only less computational costs are required. Extensive numerical experiments of the coupled nonlinear wave equation with local/nonlocal diffusion operators are provided to illustrate the energy conservation law and the computational efficiency of the schemes in long-time simulations.