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For non linear system of hyperbolic PDE, The finite volume methods work well (because of inherent conservation). Godunov scheme is a very elegant solution philosophy. For linear system, it is nothing but upwinding, but for nonlinear system, there are entropy issues. Also the Riemann solution at the local Riemann problem can not be determined if the global solution is the centered expansion fan. For this issue, we apply entropy fix. (or local linearisation e.g. Roe's) There are lot of solutions provided for this. Which one gives most consistent and accurate results?

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  • $\begingroup$ In numerical schemes for conservation laws, the conservation property is due to the use of flux-differencing. It has nothing to do with the use of finite volumes; finite difference flux-differencing methods for conservation laws are also conservative. $\endgroup$
    – David Ketcheson
    Jun 4, 2012 at 18:02
  • $\begingroup$ As Jed says, you don't need an entropy fix for Godunov's method. Even if you replace the exact Riemann solver with an approximate one, any first-order scheme will typically have so much diffusion that no entropy fix is needed. $\endgroup$
    – David Ketcheson
    Jun 4, 2012 at 18:04
  • $\begingroup$ Yes, I today I carefully read RLeVeque's and Toro's books on hyperbolic PDE and numerical methods. I was definitely wrong in my understanding before. @Jed Brown and David Ketcheson, thank you very much. @ myself, I should ask more mature questions which make sense. Thanks again. $\endgroup$
    – Subodh
    Jun 4, 2012 at 18:07
  • $\begingroup$ There was nothing wrong with your question, and it's fine to ask basic questions here. But before your question you gave an explanation, and it's true that that part wasn't correct. Perhaps you could revise the question now that you understand, in order to make it more useful. $\endgroup$
    – David Ketcheson
    Jun 4, 2012 at 18:13

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Godunov's method has an exact Riemann solver so no entropy fix is needed. A Roe solver (of which there are a few variants) uses a local linearization which has no diffusion to "fill in" the rarefaction fan, so it needs an entropy fix. Other approximate Riemann solvers, including Lax-Friedrichs, Rusanov, and the HLL family are inherently diffusive and do not need an entropy fix. Not surprisingly, an exact Riemann solver will give the best results for a given spatial discretization, but may be much more computationally expensive. Approximate Riemann solvers are more attractive with high order methods like WENO. Two books that you may find useful are

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I research about numerical solution of conservation equation, what is the newest issue about numerical solution of this equation

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    $\begingroup$ Hi user9464, and welcome to scicomp! The "answer" section is meant to address the OP's question, not to open up tangential discussions. While your question is interesting, it is best to pose it as a separate question with more detail. $\endgroup$
    – Paul
    Jun 15, 2014 at 18:18

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