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 , which is the forerunner for all other Godunov-type schemes. A second-order extension was introduced by van Leer : 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., . The Piecewise-Parabolic Method (PPM) of Colella-Woodward  and respectively, ENO schemes of Harten et.al. , 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, , offer simple and fast upwind schemes for multidimensional systems, based on an alternative positivity principle.