An Energy-Preserving and Symmetric Scheme for Nonlinear Hamiltonian Wave Equations

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