The ADER path to high-order Godunov methods

Sixty years ago, Godunov introduced his method to solve the Euler equations of gas dynamics, thus creating the Godunov school of thought for numerical approximation of hyperbolic conservation laws. The building block of the original first-order Godunov upwind method is the piece-wise constant data Riemann problem. Modern computational science, however, requires high-order accurate schemes; these may be orders-of-magnitude cheaper than first-order methods for attaining a prescribed error, and are thus mandatory in ambitious scientific and technological applications. Here we briefly review the ADER approach to construct one-step, fully discrete high-order Godunov methods for solving hyperbolic balance laws, whose building block is now the generalized Riemann problem, a piece-wise smooth data Cauchy problem including stiff or non-stiff source terms.