Energy and quadratic invariants preserving (EQUIP) multi-symplectic methods for Hamiltonian wave equations

Abstract

It is well-known that a numerical method which is at the same time geometric structure-preserving and physical property-preserving cannot exist in general for Hamiltonian partial differential equations. Motivated by EQUIP methods proposed in Brugnano et al. (2012) [13], in this paper, we present a novel class of parametric multi-symplectic Runge-Kutta methods, called EQUIP multi-symplectic methods, for Hamiltonian wave equations, which can also conserve energy simultaneously in a weaker sense with a suitable parameter. The existence of such a parameter, which enforces the energy-preserving property, is proved under certain assumptions on the fixed step sizes and the fixed initial condition. We compare the proposed method with the classical multi-symplectic Runge-Kutta method in numerical experiments, which shows the remarkable energy-preserving property of the proposed method and illustrates the validity of theoretical results. These theoretical and numerical results show that EQUIP methods can be well adapted to handle Hamiltonian partial differential equations.