I am studying the WENO scheme as presented in the D S Balsara and C-W Shu paper: "Mononicity Preserving Essentially Non-oscillatory Schemes with Increasingly High Order Accuracy", JCP, 160, (2000).

Most of the ideas are fairly straight forward, and I have been able to successfully implement various order schemes in Matlab that work fine for single 1D hypebolic problems and also that solve the Euler Equations.

My question is: How are the smoothness indicator coefficients calculated. I am obviously missing something and would be interested in a step-by-step explanation.

Any help would be much appreciated.

I'm doing my MSc. thesis on WENO and dealing with it for a long period of time, but I haven't dealed with MPWENO. I skimmed this paper, this variaton of WENO uses the same procedure to do the reconstruction in general.

In order to find nonlinear weights assigned for the candidate stencils -which are parts of your high order bigger stencil-, linear (optimal) weights and the smoothness indicators for these candidates must be found beforehand.

For uniform grids they found in a straightforward way. Optimal weights are some numeric constants, and the smoothness indicators are some combination of the values of the cells which are forming these candidate stencils. The formulations are given at the corresponding papers of Shu.

So, you use the values of the cells which are a part of the candidates and find smoothness indicators for each of these candidates, then assign nonlinear weights for them with the help of these indicators and optimal weights, and finally use these candidates to form the higher order reconstruction for your bigger stencil.

Hi Saygin,

Thank you for your response. If you have got to grips with WENO, then implementation of MPWENO and ACM are fairly minor steps. Yes they do use approximately the same calculation for the Smootness Indicators (SI).

However, my question is in relation to the actual step-by-step calculations for the basic WENO SI'. Have you managed to perform the actual calculations? If so, I would be grateful if you could outline the steps, give me a reference, or email me an attachment that details the calculations.

Regards

Hi Graham,

The actual calculations of these reconstruction polynomials, linear weights and smoothness indicators are not so easy to do. I did some calculations in order to extend my uniform domain to a non-uniform one. I can say that if you won't use rectangular domain these calculations are useless. Because in order to handle curvilinear grids coordinate transformation has to be done, and this transformation projects your domain to a uniform one.

About your question, in order to find those equations that you see in the papers, first of all you have to find Lagrange interpolation polynomials for the order you are using. For example, for 5th order WENO you have to expand there 3rd order polynomials. After that the smoothness indicators can be found by calculating the L2-norm of all derivatives of these there 3rd order polynomials. Have a look at this paper, it will help you I think. In addition don't forget to use square of the derivatives, it is not printed on the paper.

"Grid adaptation with WENO schemes for non-uniform grids to solve convection-dominated partial differential equations" J.Smit et al. Chemical Engineering Science 60 (2005) 2609-2619

Hi Saygin,

Thanks for the information. I have obtained a copy of J Smit's paper and it looks as if it may give me the information I need.

Regards