I have three questions regarding WENO schemes

1) How to actually compute the smoothness indicators $\beta_j$ for required order of polynomial? Any reference which explains the algorithm will be helpful. The formula for $\beta_j$ is given by equation (2.61) of this paper.

2) How to obtain modified wave-number plots (as given in this paper) for a WENO scheme? (For finite difference schemes with fixed stencil, the same can be obtained by taking Fourier transform of numerical approximation and equating that with the Fourier transform of exact terms). How to carry out the same when the the contribution of candidate stencils is not constant but is a function of their smoothness? (which is the case for WENO).

3) How to find dispersion errors for WENO scheme? How to carry out the analysis? (this is closely related to question # 2)

Thank you in advance.


1) If you want code, use Matthew Emmett's PyWENO package. Otherwise, the reference you give is the answer. You just need to do some simple calculus; you won't find that written down in a paper.

2) That kind of analysis does not apply directly to nonlinear schemes like WENO. If you can come up with a meaningful generalization of it, you should publish your results.

3) See #2. But also see this paper for some insight.

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