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
Contents
Quasi-linear Hyperbolic system
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
Applying the chain rule to the second term we get the quasi-linear hyperbolic system
where
The Roe Matrix
The Roe method consists of finding a matrix
Phil Roe introduced a method of parameter vectors to find such a matrix for some systems of conservation laws.
The Intercell Flux
Once the Roe matrix corresponding to the interface between two cells is found, the intercell flux is given by solving the quasi-linear system as a truly linear system.