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.