Recent Advances in the Numerical Solution of Hamiltonian Partial Differential Equations

In this paper, we study recent results in the numerical solution of Hamiltonian partial differential equations (PDEs), by means of energy-conserving methods in the class of Line Integral Methods, in particular, the Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). We show that the use of energy-conserving methods, able to conserve a discrete counterpart of the Hamiltonian functional (which derives from a proper space semi-discretization), confers more robustness to the numerical solution of such problems.