(Host: C. Garcia-Cervera)

"We present a relaxation system for the Euler equations and for ideal MHD, from which one may derive approximate Riemann solvers. The solvers satisfy a discrete entropy inequality, and preserve positivity of density and pressure under a subcharacteristic condition. Next we consider the practical implementation, and derive explicit wave speed estimates satisfying the stability conditions. We put this into an astrophysical application by comparing our new positive and entropy stable approximate Riemann solver with state-of the-art algorithms for astrophysical fluid dynamics. This is joint work with F. Bouchut and K. Waagan."