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The Roe approximate Riemann solver, devised by Phil Roe, is an approximate
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. Definition Generally speaking, Riemann solvers are specific methods for computi ...
based on the Godunov scheme and involves finding an estimate for the intercell numerical flux or Godunov flux F_ at the interface between two computational cells U_ and U_, on some discretised space-time computational domain.


Roe scheme


Quasi-linear hyperbolic system

A non-linear system of
hyperbolic partial differential equations In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n-1 derivatives. More precisely, the Cauchy problem can be ...
representing a set of
conservation laws In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, ...
in one spatial dimension can be written in the form : \frac + \frac = 0. Applying the
chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
to the second term we get the quasi-linear hyperbolic system : \frac + A(\boldsymbol)\frac = 0, where A is the Jacobian matrix of the flux vector \boldsymbol(\boldsymbol).


Roe matrix

The Roe method consists of finding a matrix \tilde(\boldsymbol_i,\boldsymbol_) that is assumed constant between two cells. The
Riemann problem A Riemann problem, named after Bernhard Riemann, is a specific initial value problem composed of a conservation equation together with piecewise constant initial data which has a single discontinuity in the domain of interest. 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 In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...
with real eigenvalues: ensures that the new linear system is truly hyperbolic. * Consistency with the exact jacobian: when \boldsymbol_i,\boldsymbol_ \rightarrow \boldsymbol we demand that \tilde(\boldsymbol_i,\boldsymbol_) = A(\boldsymbol) * Conserving \boldsymbol_-\boldsymbol_ = \tilde(\boldsymbol_-\boldsymbol_) Phil Roe introduced a method of parameter vectors to find such a matrix for some systems of conservation laws.


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.


See also

*
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. Definition Generally speaking, Riemann solvers are specific methods for computi ...


References

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Further reading

* Toro, E. F. (1999), ''Riemann Solvers and Numerical Methods for Fluid Dynamics'', Springer-Verlag. Numerical differential equations Conservation equations