Energy-preserving and symmetric fourth-order modified AVF Fourier pseudo-spectral methods for nonlinear Hamiltonian wave equations

The main objective of this work is to construct a high-order energy-preserving and symmetric fully discrete scheme, which is built upon the application of the fourth-order modified average vector field (MAVF) time integrator and the Fourier pseudo-spectral spatial approximation, for solving the nonlinear Hamiltonian wave equation. Initially, we reformulate the nonlinear Hamiltonian wave equation as an abstract infinite-dimensional separable Hamiltonian ODE system. The energy-preserving and symmetric time-stepping scheme is derived by using the fourth-order modified average vector field integrator. Subsequently, by applying the Fourier pseudo-spectral method to discretize the spatial derivatives of the second-order nonlinear Hamiltonian wave equation, we propose a fully discrete scheme that preserves energy and exhibits temporal reversibility. The rigorous error analysis demonstrates that the temporal accuracy of the proposed scheme achieves O(Δt4) under the low regularity assumption u∈C3([t0,T],Hpm0(Ω)). Finally, the numerical examples are presented to verify the accuracy, efficiency and energy conservation in long-time computation.