Space-Time Discretization of Hyperbolic Equations with Variable Collocation Points

In the field of scalar hyperbolic equations, we take into consideration the idea of building numerical schemes using two grids: the first one to represent the solution and the second one to collocate the equation. This approach is based on the experience already gained in the approximation of boundaryvalue elliptic-type problems (see for instance [1]), as well as in the field of functional ot integral-type equations (see [2]). In order to show that the same approach can be used with success also for time-dependent problems, we study finite-difference approximations of firstorder scalar hyperbolic equation in one space dimension. The representation grid is the usual uniform grid in the space-time plane. The discrete values of the solution are then assumed to be computed on a six-points stencil of such a representation grid. The approximating equations are deduced after collocation at a new point inside the stencil. It turns out that the possibility of varying the collocation point, gives a lot of freedom in the construction of the approximation method. First of all, this allows for the rediscovery of old methods and their critical analysis from a different point of view. Secondly, we have now the chance, by establishing a suitable relation between the