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

Abstract

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, the relation implies that multi-sympletic RungeKutta methods applied to equations with appropriate boundary conditions can preserve the global norm conservation and the global charge conservation respectively.

Energy/dissipation-preserving Birkhoffian multi-symplectic methods for Maxwell's equations with dissipation terms

“…which reduces to the helicity conservation law 18 and 40, and preserves the global conservation law (41). When σ 1 = σ 2 , it also preserves the local conservation laws (8) and 14, and, with appropriate boundary conditions, it preserves the global energy conservation law (10) and (15).…”

“…Applying the inverse transformation of (20) to (25)-(28), we can recover the local conservation laws (17), (19), (8) and (14).…”

“…In addition, the Hamiltonian schemes may also preserve conservation laws, dispersion relations, and so on. For instance, the multi-symplectic partitioned Runge-Kutta (MSPRK) method (see, e.g., [7,8], and references therein ), which is one of the most important and popular class of Hamiltonian schemes, preserves quadratic conservation laws for autonomous Hamiltonian partial differential equations (PDEs). In references [9,10,11,12], Hamiltonian symplectic/multi-symplectic schemes are constructed for Maxwell's equations in free space and energy-conserving properties of such structure-preserving schemes for Maxwell's equations are investigated.…”

Symplectic and multisymplectic numerical methods for Maxwell’s equations

“…From the multisymplectic literature, (13) is a quadratic conservation law according to [33] while (14) and (15) are the multisymplectic energy and momentum conservation laws with S(z) = 0. The proposition for (13) holds from a result in [33], while (14) and (15) hold by generalization to higher spatial dimensions a theorem in [16]. h From Proposition 3.2, we have the following corollary for the boxscheme (25).…”

“…In this section, we follow the general dispersion analysis for numerical PDEs in [36] to investigate the dispersion relations for the four methods: symplectic method (18), multisymplectic boxscheme (25), multisymplectic leapfrog method (30) and the multisymplectic Yee's method (33) and (34), etc.…”