We want to solve the hyperbolic system of conservation laws

by Godunov-type schemes. To this end we proceed in two steps. First, we
introduce a small spatial scale, , and we consider the corresponding
(Steklov) sliding average of ,

The sliding average of (1.2.1) then yields

Next, we introduce a small time-step, , and integrate over the
slab ,

We end up with an equivalent reformulation of the conservation law
(1.2.1): it expresses the precise relation between the sliding
averages, , and their underlying pointvalues,
. We shall use this reformulation, (1.2.3), as the
starting point for the construction of Godunov-type schemes.

We construct an approximate solution, , at the discrete
time-levels, . Here, is a piecewise polynomial
written in the form

where are algebraic polynomials supported at the discrete cells,
, centered around the midpoints, .
An *exact* evolution of based on (1.2.3), reads

To construct a Godunov-type scheme, we *realize* (1.2.4) -- or at
least an accurate approximation of it, at discrete gridpoints. Here, we
distinguish between the main methods, according to their way of
*sampling* (1.2.4): these two main sampling methods
correspond to upwind schemes and central schemes.

Mon Dec 8 17:34:34 PST 1997