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I have been following Toro's book on Riemann solvers to implement a finite-volume scheme for computational fluid dynamics. The Riemann solvers presented in the book seem to be fairly tightly coupled to the equations in question. There are papers on how to construct the schemes for various sets of equations such as the Euler equations, relativistic hydrodynamics, or magneto-hydrodynamics (for example).

I am wondering: is there prior work on a more general formulation of a scheme (like HLLC or Roe) that solves the Riemann problem for a larger class of equations instead of constructing it case-by-case?

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I cant think of any truly generic ones except the Lax-Friedrichs flux, which is very dissipative. The next simplest would be the Rusanov flux which needs knowledge of minimum and maximum speeds arising in the Riemann problem, which of course depends on the particular form of the flux function. While this is also called a central flux, it can be interpreted as a Riemann solver. Most other more sophisticated solvers need knowledge of other intermediate speeds. The HLL type solvers require solving jump conditions to find intermediate states which depends on the flux. Still more sophisticated ones like Roe need knowledge of eigenvectors also.

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  • $\begingroup$ Thank you for elaborating. In the meantime, I found a paper by Castro & Toro (2014) titled "Roe-type Riemann solvers for general hyperbolic systems". It appears to generalize the Roe formalism somewhat. As you rightly point out though, it still requires knowledge of the eigenstructure of the system. $\endgroup$
    – Emilia Bopp
    Commented Oct 9, 2017 at 14:30

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