Numerical solutions of Hamiltonian PDEs: a multi-symplectic integrator in light-cone coordinates

Hugo Ricateau (1) , Leticia F Cugliandolo (1)

1 LPTHE - Laboratoire de Physique Théorique et Hautes Energies

We introduce a novel numerical method to integrate partial differential equations representing the Hamiltonian dynamics of field theories. It is a multi-symplectic integrator that locally conserves the stress-energy tensor with an excellent precision over very long periods. Its major advantage is that it is extremely simple (it is basically a centered box scheme) while remaining locally well defined. We put it to the test in the case of the non-linear wave equation (with quartic potential) in one spatial dimension, and we explain how to implement it in higher dimensions. A formal geometric presentation of the multi-symplectic structure is also given as well as a technical trick allowing to solve the degeneracy problem that potentially accompanies the multi-symplectic structure.

Domaines: Analyse numérique [math.NA]; Physique mathématique [math-ph]; Physique Numérique [physics.comp-ph]; Formation de Structures et Solitons [nlin.PS]

Hugo Ricateau, Leticia F Cugliandolo. Numerical solutions of Hamiltonian PDEs: a multi-symplectic integrator in light-cone coordinates. 2017. ⟨hal-01474871⟩