Riemann solvers are the core computation in some numerical methods for hyperbolic conservation laws. For such methods, the Riemann solvers are one of the most computationally demanding part of the entire algorithm. For this reason a common strategy is to develop efficient methods that approximate the solution of a Riemann problem.

In recent years some highly efficient iterative algorithms have been proposed to approximate the solution of Riemann problems to arbitrary precision.

This suggests that one could obtain the exact solution of Riemann problems (up to machine precision) in few iterations.

As a result, one could attempt to consider the exact solution of Riemann problems instead of the more standard approximations without an excessive over head on computational cost.

In this project we are interested on using state of the art exact Riemann solvers within the context of high-order finite volume methods and compare the overall quality, computational cost and robustness of the solution versus more standard approximate Riemann solvers. To do this we plan to concentrate on the Euler equations and the shallow

water model.