Abstract
The Godunov theorem provides a limitation for finite volume schemes, linear and monotone; only first
order of accuracy can be achieved. A family of non-linear schemes, called ADER (Arbitrary accuracy
DERivative Riemann problems), attempting to circumvent Godunov’s theorem, has been proposed.
Building block for these schemes are: the polynomial reconstruction of cell averages and; the solution
of special Cauchy problem, the so called generalized Riemann problems. The reconstruction and the
solution of Generalized Riemann problems allow us to construct schemes of accuracy p > 1 such that
spurious oscillations near high gradients are controlled. Advantages of these finite volume schemes
are; efficiency, stability and accuracy.
In this seminar are presented: some notions on hyperbolic problems and elementary waves; classical
Riemann problems and Godunov’s type schemes; the Godunov theorem and the Harten theorem, which
are the main results regarding the construction of numerical schemes; the generalized Riemann
problems and the construction of high-order finite volume methods.