An energy-preserving and symmetric scheme for nonlinear Hamiltonian wave equations

In this work, we derive and analyze a novel energy-preserving and symmetric scheme for nonlinear Hamiltonian wave equations, which can exactly preserve the energy of the underlying wave equations. To this end, we first define and discuss the bounded operator-valued functions on the underlying domain. We then introduce an operator-variation-of-constants formula, and based on which we present a new energy-preserving scheme for the nonlinear Hamiltonian wave equations. The scheme preserves the energy of the original continuous Hamiltonian system exactly. Compared with the existing work on this topic, such as the well-known Average Vector Field (AVF) formula for Hamiltonian ordinary differential equations, the new energy-preserving scheme avoids the semi-discretization of spacial derivatives and exactly preserves the Hamiltonian of the original continuous Hamiltonian wave equation. This point is greatly significant in contrast with the AVF formula, since the AVF formula can preserve only the energy of Hamiltonian ordinary differential equations. Hence, the main theme of this paper is to establish a new scheme which can exactly preserve the energy of the nonlinear Hamiltonian wave equation. The paper is also accompanied by some examples to illustrate our results to some extent.