The Derivative Riemann Problem: The basis for high order ADER Schemes

The corner stone of arbitrary high order schemes (ADER schemes) is the solution of the derivative Riemann problem at the element interfaces, a generalization of the classical Riemann problem first used by Godunov in 1959 to construct a first-order upwind numerical method for hyperbolic systems. The derivative Riemann problem extends the possible initial conditions to piecewise smooth functions, separated by a discontinuity at the interface. In the finite volume framework, these piecewise smooth functions are obtained from cell averages by a high order non-oscillatory WENO reconstruction, allowing hence the construction of non-oscillatory methods with uniform high order of accuracy in space and time