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 ,,, , the magneto-hydrodynamics equations , viscoelastic flows| hyperbolic systems with relaxation source terms ,, non-linear optics ,, and slow moving shocks .
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 , 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  (see also ). It was extended to higher-order of accuracy as well as for more space dimensions (consult , , ,  and , for the two-dimensional case, and , ,  and  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 ,. Higher orders are treated in .
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.