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I'm reading now

Shu, C.-W. (1999). High Order ENO and WENO Schemes for Computational Fluid >Dynamics. (T. J. Barth & H. Deconinck, Eds.)Lecture Notes in Computational >Science and Engineering, 9, 439–582. doi:10.1007/978-3-662-03882-6_5

The author provides two formulations of WENO methods: finite-difference and finite-volume.

What I don't understand is why in finite-volume formulation we use Riemann solver, but in finite-difference we don't use it?

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They are different methods and presence of a Riemann solver is the distinguishing characteristic.

Finite Difference WENO starts with vertex values, evaluates the fluxes associated with left- and right-traveling waves, and uses weighted averaging to define a flux at staggered points. (In the paper, the average is done in local characteristic variables.) FD WENO is inexpensive, but only works on smooth meshes and requires a flux splitting.

Finite Volume WENO reconstructs a piecewise discontinuous state and evaluates it at quadrature points on cell interfaces. A single-valued flux is evaluated using the multivalued state at those quadrature points. FV WENO is more general, but Riemann solvers are often more expensive and most full-accuracy variants end up needing many more evaluations in multiple directions (due to the need for transverse reconstruction). See Zhang, Zhang, and Shu, On the Order of Accuracy and Numerical Performance of Two Classes of Finite Volume WENO Schemes, 2011 for more details.

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  • $\begingroup$ To add to this answer, the OP should know that this is the fundamental difference between finite difference and finite volume methods, and doesn't have anything to do with ENO/WENO. $\endgroup$ – Aurelius Nov 12 '13 at 18:21
  • $\begingroup$ @Jed, thank you for the answer and the link! $\endgroup$ – Dmitry Kabanov Nov 12 '13 at 18:41
  • $\begingroup$ @Aurelius, could you please point me to some literature source to read about this difference? $\endgroup$ – Dmitry Kabanov Nov 12 '13 at 18:42
  • $\begingroup$ @KabanovDmitry, I'd have to just point you to the wiki page on the FV method. The core difference between FV and FD is that FV works in an integral form, and fluxes are evaluated by surface integrals. In general on an FV cell surface the reconstructions are discontinuous on either side, so a Riemann solver is needed to compute a unique flux between the two sides. $\endgroup$ – Aurelius Nov 12 '13 at 18:57

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