The correct use of the Lax-Friedrichs method

are concerned with the structure of the operator corresponding to the Lax-Friedrichs method. At first, the phenomenae which may arise by the naive use of the Lax-Friedrichs scheme are analyzed. In particular, it turns out that the correct definition of the method has to include the details of the discretization of the initial condition and the computational domain. Based on the results of the discussion, we give a recipe that ensures that the number of extrema within the discretized version of the initial data cannot increase by the application of the scheme. The usefulness of the recipe is confirmed by numerical tests.