Note that the polynomial pieces of are supported in the cells, , with interfacing breakpoints at the half-integers gridpoints, .

We recall that upwind schemes (1.2.5) were based on sampling
(1.2.4) in the *midcells*, . In contrast, central
schemes are based on sampling (1.2.8) at the *interfacing breakpoints*,
, which yields

We want to utilize (1.2.9) in terms of the known cell averages at time
level . The remaining task is therefore
to recover the *pointvalues*
,
and in particular, the *staggered averages*, . As before, this task is accomplished in two main steps:

- First, we use the given cell averages , to
*reconstruct*the pointvalues of as piecewise polynomial approximation

In particular, the staggered averages on the right of (1.2.9) are given by

The resulting central scheme (1.2.9) then reads

- Second, we follow the
*evolution*of the pointvalues along the mid-cells, , which are governed by

Let denote the eigenvalues of the Jacobian . By hyperbolicity, information regarding the interfacing discontinuities at

propagates no faster than . Hence, the mid-cells values governed by (1.2.13), , remain free of discontinuities, at least for sufficiently small time step dictated by the CFL condition . Consequently, since the numerical fluxes on the right of (1.2.12), , involve only smooth integrands, they can be computed within any degree of desired accuracy by an appropriate quadrature rule.

**Figure 1.2.2:** *Central differencing by Godunov-type scheme.*

It is the *staggered* averaging over the fan of left-going
and right-going waves centered at the half-integered interfaces,
, which characterizes the *central*
differencing, consult Figure 1.2.2.
A main feature of these central schemes - in contrast to upwind ones, is the
computation of *smooth* numerical fluxes along the mid-cells,
, which avoids the costly (approximate)
Riemann solvers. A couple of examples of central Godunov-type
schemes is in order.

The first-order Lax-Friedrichs (LxF) approximation
is the forerunner for such central schemes -- it is based on
piecewise constant reconstruction,
with . The resulting central scheme, (1.2.12),
then reads
(with the usual fixed mesh ratio )

Our main focus in the rest of this chapter is on non-oscillatory higher-order
extensions of the LxF schemes.

Mon Dec 8 17:34:34 PST 1997