An implicit, compact, finite difference method to solve hyperbolic equations

At present most approximate (discrete) solutions of time dependent hyperbolic equations are obtained by explicit finite difference methods, where the maximal allowable time step is given by a condition of numerical stability (i.e., the CFL condition). This report contains the development and the analysis of an implicit method of high order accuracy which is unconditionally stable, thus allowing to progress much faster in time. Furthermore, the presence of additional artificial boundary conditions does have an influence on the accuracy of the solution but not on the stability of the method. The numerical scheme has been checked considering two examples: the solution of the (linear) wave equation and the non linear Euler equations of fluid dynamics.