Formally, the Godunov scheme and all its high-order accurate extensions can be written as follows:
and the application in sequence of the three operators 
(reconstruction),
(time-evolution), and 
(cell-average) allows the
updating of the cell-averages solution from the initial time 
to
the final time 
.
All the schemes which can be written in this way are generalisations of
the original Godunov scheme. In fact, if 
is a first-order accurate
piecewise-constant reconstruction,
Eq. (11) yields exactly the original first-order accurate Godunov scheme; if R is a second-order accurate piecewise-linear reconstruction
such that
then Eq. (11) is the abstract form of the second-order extensions of the Godunov scheme described by van Leer (1979), [16].
The major disadvantage of this approach is the complexity introduced into a difference scheme through a Riemann Solver and the cost in terms of number of operations and CPU-time. The simplest such scheme, Godunov first-order method, requires about twice as much computer time per cell per time step as the second-order MacCormack scheme with an artificial viscosity. But when the flow is discontinous the Riemann solver yields a more reliable result than a calculation based upon a smooth flow model, see Woodward and Colella (1984) [17].
