Numerical solutions of Hamiltonian PDEs: a multi-symplectic integrator in light-cone coordinates
For researchers in numerical methods for field theories, this provides a simple yet locally well-defined integrator that conserves the stress-energy tensor, though it is an incremental improvement over existing multi-symplectic schemes.
The paper introduces a novel multi-symplectic integrator for Hamiltonian PDEs that locally conserves the stress-energy tensor with excellent precision over long periods, demonstrated on the nonlinear wave equation in 1D.
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.