Quadratic invariants and multi-symplecticity of partitioned Runge-Kutta methods for Hamiltonian PDEs

In this paper, we study the preservation of quadratic conservation laws of Runge-Kutta methods and partitioned Runge-Kutta methods for Hamiltonian PDEs and establish the relation between multi-symplecticity of Runge-Kutta method and its quadratic conservation laws. For Schrödinger equations and Dirac equations, it reveals that multi-symplectic Runge-Kutta methods applied to equations with appropriate boundary conditions can preserve the global norm conservation and the global charge conservation, respectively.