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:
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),
(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.