Roe solver

The Roe approximate Riemann solver, devised by Phil Roe, is an approximate Riemann solver based on the Godunov scheme and involves finding an estimate for the intercell numerical flux or Godunov flux ${\displaystyle F_{i+{\frac {1}{2}}}}$ at the interface between two computational cells ${\displaystyle U_{i}}$ and ${\displaystyle U_{i+1}}$, on some discretised space-time computational domain.

A non-linear system of hyperbolic partial differential equations representing a set of conservation laws in one spatial dimension can be written in the form

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