Symplectic Hamiltonian Finite Element Methods for Semilinear Wave Propagation

This paper presents Hamiltonian finite element methods for approximating semilinear wave propagation problems, including the nonlinear Klein–Gordon and sine-Gordon equations. The aim is to obtain accurate high-order approximations while conserving physical quantities of interest such as energy. To achieve conservation properties at a discrete level, we propose semidiscrete schemes based on two Hamiltonian structures of the equation. These include Mixed finite element methods, discontinuous Galerkin methods, and hybridizable discontinuous Galerkin methods (HDG). In particular, we propose a new class of DG methods using time operators to define the numerical traces, ultimately leading to an energy-conserving scheme. Time discretization uses Symplectic explicit-partitioned and diagonally-implicit Runge–Kutta schemes. Furthermore, the paper showcases several numerical examples that demonstrate the accuracy and energy conservation properties of the approximations, along with the simulation of soliton cloning.