The correct formulation of numerical models for free-surface hydrodynamics often requires the solution of special linear systems whose coefficient matrix is a piecewise constant function of the solution itself. In so doing one may prevent the development of unrealistic negative water depths. The resulting piecewise linear systems are equivalent to particular linear complementarity problems whose solutions could be obtained by using, for example, interior point methods. These methods may have a favorable convergence property, but they are purely iterative and convergence to the exact solution is proven only in the limit of an infinite number of iterations. In the present paper a simple Newton-type procedure for certain piecewise linear systems is derived and discussed. This procedure is shown to have a finite termination property, i.e., it converges to the exact solution in a finite number of steps, and, actually, it converges very quickly, as confirmed by a few numerical tests.

We introduce a new family of symplectic integrators for canonical Hamiltonian systems. Each method in the family depends on a real parameter $\alpha$. When $\alpha=0$ we obtain the classical Gauss collocation formula of order $2s$, where $s$ denotes the number of the internal stages. For any given non-null $\alpha$, the corresponding method remains symplectic and has order $2s-2$; hence it may be interpreted as an $O(h^{2s-2})$ (symplectic) perturbation of the Gauss method. Under suitable assumptions, we show that the parameter $\alpha$ may be properly tuned, at each step of the integration procedure, so as to guarantee energy conservation in the numerical solution, as well as to maintain the original order $2s$ as the generating Gauss formula.