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.