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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. A Riemann solver is a numerical method used to solve a Riemann problem. They are heavily used in computational fluid dynamics and computational magnetohydrodynamics.Godunov is credited with introducing the first exact Riemann solver for the Euler equations, by extending the previous CIR (Courant-Isaacson-Reeves) method to non-linear systems of hyperbolic conservation laws. Modern solvers are able to simulate relativistic effects and magnetic fields. For the hydrodynamic case latest research results showed the possibility to avoid the iterations to calculate the exact solution for the Euler equations.