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

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