Here, it remains to recover the

- First, the
*reconstruction*- we recover the pointwise values of at , by a reconstruction of a piecewise polynomial approximation

- Second, the
*evolution*-- are determined as the solutions of the generalized Riemann problems

**Figure 1.2.1:** *Upwind differencing by Godunov-type scheme.*

The original Godunov scheme is based on piecewise-constant reconstruction,
, followed by an exact Riemann solver.
This results in a first-order accurate upwind method [11],
which is the forerunner for all other Godunov-type schemes.
A second-order extension was introduced
by van Leer [19]:
his MUSCL scheme reconstructs a piecewise linear
approximation, , with linear pieces of
the form
so that . Here the -s are possibly
limited slopes which are reconstructed from the known cell-averages,
.
(Throughout this lecture we use primes, ,
to denote *discrete* derivatives, which approximate the corresponding
differential ones).
A whole library of limiters is available in this context,
so that the co-monotonicity of with
is guaranteed, e.g., [42].
The Piecewise-Parabolic Method (PPM)
of Colella-Woodward [6] and
respectively, ENO schemes of Harten et.al. [13], offer, respectively,
third- and higher-order Godunov-type upwind schemes.
(A detailed account of ENO schemes can be found in lectures
of C.W. Shu in this volume).
Finally, we should not give the impression that limiters are used
exclusively in conjunction with Godunov-type schemes. The
*positive schemes* of Liu and Lax, [27],
offer simple and fast upwind schemes for multidimensional systems,
based on an alternative positivity principle.

Mon Dec 8 17:34:34 PST 1997