I am currently trying to understand what WENO schemes are and most of the literature on web talks about cell-face reconstruction. What I am unable to understand is the origin of these discontinuities at the cell-faces. Can anyone explain where exactly does a WENO scheme come into picture and why is it used?
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In numerical solution of differential equations, WENO (weighted essentially non-oscillatory) methods are classes of high-resolution schemes. WENO are used in the numerical solution of hyperbolic partial differential equations.
as you can find in the wikipedia link. The derivation of the weno scheme can be found in the original article. The fact that they are used near discontinuities is because, usually, finite volume (but also finite elements) schemes, in proximity of the discontinuity present spurious oscillation, also known Gibbs phenomenon. For this reason, usually. near the discontinuities you need to use very refined cells and with linear schemes you can achieve only 1st order accuracy to obtain a non-oscillatory behaviour (have a look at this link). To obtain non-oscillatory schemes with higher than first order accuracy, one must consider nonlinear schemes. In this sense the TVD schemes revolutioned the fluid mechanics but still they have a drawback: they have degeneracy of accuracy to first order near smooth extrema. For this reason the ENO and WENO schemes were introduced. For what it concerns the discontinuities they can appear:
- in monophase flows in case of shock waves (you can think at it in case of the whole domain or on a single discretized cell)
- in multiphase flow in case of discontinuity between the two phases (in this case you can have a look at the chapter 2 here, where are explained the different possible formulations of multiphase flow.
