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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In numerical analysis and computational fluid dynamics, Godunov''s scheme is a conservative numerical scheme, suggested by S. K. Godunov in 1959, for solving partial differential equations. In this method, the conservative variables are considered as piecewise constant over the mesh cells at each time step and the time evolution is determined by the exact solution of the Riemann problem (shock tube) at the inter-cell boundaries (Hirsch, 1990).