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.