We discuss the occurrence of oscillations
when using central schemes of the Lax-Friedrichs type (LFt), Rusanov's method and the staggered and
non-staggered second order Nessyahu-Tadmor (NT) schemes.
Although these schemes are monotone or TVD, respectively,
oscillations may be introduced at local data extrema.
The dependence of oscillatory properties on the numerical viscosity
coefficient is investigated rigorously for the LFt schemes, illuminating also
the properties of Rusanov's method. It turns out, that schemes with a large viscosity coefficient are
prone to oscillations at data extrema. For all LFt schemes except for the classical
Lax-Friedrichs method, occurring oscillations are damped in the course of a computation.
This damping effect also holds for Rusanov's method. Concerning the NT schemes, the non-staggered
version may yield oscillatory results, while it can be shown rigorously that the staggered NT
scheme does not produce oscillations when using the classical minmod-limiter under a
restriction on the time step size. Note that this restriction is not the
same as the condition ensuring the TVD property.
Numerical investigations of one-dimensional scalar problems and of
the system of shallow water equations in two dimensions
with respect to the phenomenon complete the paper.