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

${\displaystyle {\frac {\partial {\boldsymbol {U}}}{\partial t}}+{\frac {\partial {\boldsymbol {F}}({\boldsymbol {U}})}{\partial x}}=0.}$

Applying the chain rule to the second term we get the quasi-linear hyperbolic system

${\displaystyle {\frac {\partial {\boldsymbol {U}}}{\partial t}}+A({\boldsymbol {U}}){\frac {\partial {\boldsymbol {U}}}{\partial x}}=0,}$

where ${\displaystyle A}$ is the Jacobian matrix of the flux vector ${\displaystyle {\boldsymbol {F}}({\boldsymbol {U}})}$.

Roe matrix

The Roe method consists of finding a matrix ${\displaystyle {\tilde {A}}({\boldsymbol {U}}_{i},{\boldsymbol {U}}_{i+1})}$ that is assumed constant between two cells. The Riemann problem can then be solved as a truly linear hyperbolic system at each cell interface. The Roe matrix must obey the following conditions:

• Diagonalizable with real eigenvalues: ensures that the new linear system is truly hyperbolic.
• Consistency with the exact jacobian: when ${\displaystyle {\boldsymbol {U}}_{i},{\boldsymbol {U}}_{i+1}\rightarrow {\boldsymbol {U}}}$ we demand that ${\displaystyle {\tilde {A}}({\boldsymbol {U}}_{i},{\boldsymbol {U}}_{i+1})=A({\boldsymbol {U}})}$
• Conserving ${\displaystyle {\boldsymbol {F}}_{i+1}-{\boldsymbol {F}}_{i}={\tilde {A}}({\boldsymbol {U}}_{i+1}-{\boldsymbol {U}}_{i})}$

Phil Roe introduced a method of parameter vectors to find such a matrix for some systems of conservation laws.^[1]

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.