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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. The Roe approximate Riemann solver, devised by Phil Roe, is an approximate Riemann solver based around the Godunov scheme and involves finding an estimate for the intercell numerical flux or Godunov flux F_{i + frac{1}{2}} at the interface between two computational cells Ui and Ui + 1, on some discretised space-time computational domain. 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[1], 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.