# Polynomial grid convergence studies

Hi Freddie,

Moderator: this was moved here from a topic about ILES in PyFR

I have a dumb question regarding this. As I understand, iLES in PyFR is achieved by approximating integration through Gaussian quadrature rule with certain quadrature degree. Does that mean doing grid convergence analysis by just playing around polynomial degrees is not very meaningful. As we increasing polynomial order, quadrature rule order increases, then we are marching to very resolved DNS eventually.

Best wishes,
Zhenyang

If you are running a simulation with anti-aliasing then as you increase the solution polynomial order you also need to increase the quadrature rule order.

However, if no anti-aliasing is enabled then increasing the polynomial order will increase the amount of resolution and hence is suitable for convergence analysis.

Regards, Freddie.

I would say that this isn’t particularly meaningful. When doing a grid convergence study, the aim is to show that the result is converging to a, hopefully true, value. Key to this is the consistency and convergence proof for the method. What we are generally looking for is on the grids tested the solution will asymptote to the true value, which for solutions under some regularity assumptions it will.

However, when you change the polynomial order in FR you are changing the ILES filter. So I suspect that it will likely lead to a solution that will eventually asymptote to the true value, again under some regularity conditions, however for the orders we are talking about I imagine it won’t give a very clear picture.

Additionally, for non-linear equations, the precision CFL condition for mixed hyperbolic-parabolic equations is not well understood. Therefore, as you change order and change dt, you will also be changing the effect of the filter caused by the time scheme. This will add a further layer of murkiness to the result.

Edit: Me and Freddie seem to disagree. What fun.

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