In recent years, central schemes for approximating solutions of hyperbolic conservation
laws, received a considerable amount of renewed attention.
A family of high-resolution, non-oscillatory, *central* schemes, was
developed to handle such problems.
Compared with the 'classical' *upwind * schemes, these *central*
schemes were shown to be both simple and
stable for a large variety of problems ranging from one-dimensional scalar problems
to multi-dimensional systems of conservation laws.
They were successfully implemented for a variety of other related problems,
such as, e.g., the incompressible Euler
equations [25],[22],[20],
[21], the magneto-hydrodynamics equations [45], viscoelastic flows|[20]
hyperbolic systems with relaxation source
terms [4],[37],[38]
non-linear optics [36],[7],
and slow moving shocks [17].

The family of high-order *central* schemes we deal with,
can be viewed as a direct extension to the
first-order, Lax-Friedrichs (LxF) scheme [9], which on one hand
is robust and stable, but on the
other hand suffers from excessive dissipation.
To address this problematic property of the LxF scheme, a Godunov-like
second-order central scheme was developed by Nessyahu and Tadmor (NT) in
[31] (see also [41]).
It was extended to higher-order of accuracy as
well as for more space dimensions (consult [1],
[16], [2], [3]
and [21], for the two-dimensional case,
and [40], [14], [29]
and [24] for the third-order schemes).

The NT scheme is based on reconstructing, in each time step, a piecewise-polynomial interpolant from the cell-averages computed in the previous time step. This interpolant is then (exactly) evolved in time, and finally, it is projected on its staggered averages, resulting with the staggered cell-averages at the next time-step. The one- and two-dimensional second-order schemes, are based on a piecewise-linear MUSCL-type reconstruction, whereas the third-order schemes are based on the non-oscillatory piecewise-parabolic reconstruction [28],[29]. Higher orders are treated in [39].

Like *upwind* schemes,
the reconstructed piecewise-polynomials used by the
central schemes, also make use
of non-linear limiters which guarantee the overall non-oscillatory nature of the
approximate solution. But unlike the upwind schemes, central schemes
avoid the intricate and time consuming Riemann solvers;
this advantage is particularly important
in the multi-dimensional setup, where no such Riemann solvers exist.

Mon Dec 8 17:34:34 PST 1997