  The Riemann problem for first-order hyperbolic systems of partial
differential equations is of fundamental importance for both theoretical and
numerical purposes. Many approximate solvers have been developed for such
systems; exact solution algorithms have received less attention because
computation of the exact solution typically requires iterative solution of
algebraic equations. Iterative algorithms may be less computationally efficient
or might fail to converge in some cases. We investigate the achievable
efficiency of robust iterative Riemann solvers for relatively simple systems,
focusing on the shallow water and Euler equations. We consider a range of
initial guesses and iterative schemes applied to an ensemble of test Riemann
problems. For the shallow water equations, we find that Newton's method with a
simple modification converges quickly and reliably. For the Euler equations we
obtain similar results; however, when the required precision is high, a
combination of Ostrowski and Newton iterations converges faster. These solvers
are slower than standard approximate solvers like Roe and HLLE, but come within
a factor of two in speed. We also provide a preliminary comparison of the
accuracy of a finite volume discretization using an exact solver versus
standard approximate solvers.
