Roe solver
Encyclopedia
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
at the interface between two computational cells 
and 
, on some discretised space-time computational domain.
can be written in the form
Applying the chain rule
the second term we get the quasi-linear hyperbolic system
where
is the jacobian matrix of the flux vector 
.
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:
Philip L. Roe
Philip L. Roe is a Professor of Aerospace Engineering at the University of Michigan in Ann Arbor. He is known for his work in the field of Computational Fluid Dynamics and Magnetohydrodynamics. Roe made fundamental contributions to the development of high-resolution schemes for hyperbolic...
, is an approximate Riemann solver
Riemann solver
A Riemann solver is a numerical method used to solve a Riemann problem. They are heavily used in computational fluid dynamics and computational magnetohydrodynamics.-Exact solvers:...
based around the Godunov scheme and involves finding an estimate for the intercell numerical flux or Godunov flux
at the interface between two computational cells and , on some discretised space-time computational domain.Quasi-linear Hyperbolic system
A non-linear system of hyperbolic partial differential equations representing a set of conservation laws in one spatial dimensioncan be written in the form
Applying the chain rule
Chain rule
In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...
the second term we get the quasi-linear hyperbolic system
where
is the jacobian matrix of the flux vector .The Roe Matrix
The Roe method consists of finding a matrix that is assumed constant between two cells. The Riemann problemRiemann problem
A Riemann problem, named after Bernhard Riemann, consists of a conservation law together with piecewise constant data having a single discontinuity. 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
we demand that 
- Conserving
Phil RoePhilip L. RoePhilip L. Roe is a Professor of Aerospace Engineering at the University of Michigan in Ann Arbor. He is known for his work in the field of Computational Fluid Dynamics and Magnetohydrodynamics. Roe made fundamental contributions to the development of high-resolution schemes for hyperbolic...
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.
Further reading
- Toro, E. F. (1999), Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer-Verlag.