A new efficient implementation for HBVMs and their application to the semilinear wave equation.

In this thesis we have provided a detailed description of the low-rank Runge-Kutta family of Hamiltonian Boundary Value Methods (HBVMs) for the numerical solution of Hamiltonian problems. In particular, we have studied in detail their main property: the conservation of polynomial Hamiltonians, which results into a practical conservation for generic suitably regular Hamiltonians. This property turns out to play a fundamental role in some problems where the error on the Hamiltonian, usually obtained even when using a symplectic method, would be not negligible to the point of affecting the dynamics of the numerical solution. The research developed in this thesis has addressed two main topics. The first one is a new procedure, based on a particular splitting of the matrix defining the method, which turns out to be more effective of the well-known blended-implementation, as well as of a classical fixed-point iteration when the problem at hand is stiff. This procedure has been applied also to second order problems with separable Hamiltonian function, resulting in a cheaper computational cost. The second topic addressed is the application of HBVMs for the full discretization of a method of lines approach to numerically solve Hamiltonian PDEs. In particular, we have considered the semilinear wave equation coupled with either periodic, Dirichlet or Neumann boundary conditions, and the application of a (practically) energy conserving HBVM method to the semi-discrete problem obtained by means of a second order finite-difference approximation in space. When the problem is coupled with periodic boundary conditions we have also considered the case of higher-order finite-difference spatial discretizations and the case when a Fourier-Galerkin method is used for the spatial semi-discretization. The proposed methods are able to provide a numerical solution such that the energy (which can be conserved or not, depending on the assigned boundary conditions) practically satisfies its prescribed variation in time. A few numerical tests for the sine-Gordon equation have given evidence that, for some problems, there is an effective advantage in using an energy-conserving method for the time integration, with respect to the use of a symplectic one. Moreover, even though HBVMs are implicit method, their computational cost for the considered problem turns out to be competitive even with respect to that of explicit solvers of the same order, which, furthermore, may suffer from stepsize restrictions due to stability reasons, whereas HBVMs are A-stable.