[Barmanissimo]

Hello everyone, I'd like to ask You for help with my issue:

Using MATLAB I'm trying to retrieve results from the article by Flyer and Wright (https://www.researchgate.net/publica...asis_functions) presented in subsection "Solid body rotation" (pp. 1066-1075). I focus on advecting Gaussian bell initial condition. Using the code and parameters such as timestep size or wind velocity presented in the paper's appendix I want to obtain spatial convergence results from Fig. 10b (computed using 4th order Runge Kutta timestepping). For coarser nodesets, it works quiet well, while for the finer ones the errors does not fall down anymore (solid line in the figure below):
flyer_conv.PNG
The situation gets a little better when I use 5th order Rung-Kutta-Fehlberg time integration and decrease the timestep size (dashed lines). Note, that - in compliance with the paper - timestep size is kept constant for all nodesets.

Looking at the figure, it seems to me, that for the denser nodesets temporal errors outweigh spatial ones and that's why errors behave this way.

Following this thought, I plotted the differentiation matrix (defined in the paper at page 1067 and 1068) eigenvalues for each nodeset to check if it falls into RK4 or RKF5 stability regions, and what I've found is that not all of them do:
flyer_eig1.PNG
And after some magnification:
flyer_eig2.PNG

What I've also noticed is that increasing the nodes count results in bigger real (especially positive) and imaginary parts of eigenvalues, which - for me - might explain the errors behaviour (bigger eigenvalues fall out of the RK stability region "more", thus introducing bigger temporal errors).

Let me also attach my plot of temporal evolution of errors for each nodeset (L2 normalized error is plotted black, Linf error - red, numbers at the arrows denote node count):
flyer_time_evolution.jpg

It looks like there's no difference in errors evolution for the two finest nodesets. I also do not have any explanation for errors nearly not increasing in time for the coarser nodesets.

Concluding, here are the things that I'd like to ask You:

1. Can the fact, that the differentiation matrix eigenvalues don't all lay exactly inside RK methods stability regions be the reason for such strange convergence behaviour, or their magnitudes are so small that it doesn't actually matter??
2. Shouldn't L2 error be always smaller than Linf? - after all L2 is in a way "mean" error, Linf is simply the maximum value of error.
3. Has anyone ever had similar problem with temporal errors outweighing temporal ones and maybe found a solution for it? The paper can't be wrong, so I suppose I'm continuously making some little mistake (especially that the code is given in the paper) I can't see and that's the reason for all this.

I'm currently working on a similar code, for solving advection equation with Gaussian bell IC on [0, 1) x [0, 1) square with periodic BCs (implemented by incorporating values of RBFs from "neighbouring" domains into my interpolation matrix [named "A" in the paper]) and I'm getting very similar results.

Thank You for Your time and help.

[FMDenaro]

The presence of the onset of a numerical instability appears clearly in the solution by the superimposed wiggles that appears at high frequency, close to the Nyquist one. You can see the plot of the solution to see if it is free of oscillations.
But I suppose that the fact that the slope does no longer diminish depends on the type of accuracy test. When you fix the time step dt and change only the spatial step h you reduce only the part of the error related to the spatial derivative while taking constant the error due to the temporal scheme. After a certain level, the constant slope appear.
You can try to see if this is the case by reducing the time-step or by doing the accuracy test at constant CFL, that is by changing dt and h.

[Barmanissimo]

Dear Filippo,

checking the numerical solution plot, there are no wiggles You've mentioned. I've also conducted tests with constant Courant number (Co = 0.2, as for the finest nodeset for constant dt test case described in the first post). For that setup, spatial convergence becomes algebraic, not spectral. I suppose it is due to the accuracy of time stepping being worse than spatial RBF interpolation (which - in theory - is spectral) so finally temporal errors outweigh spatial errors once again.

The only thing that I can't get is why this errors behaviour is not present in the paper's results. Once I've also suspected some kind of MATLAB's precision to have something to do with it, but the order of the constant-slope part of spatial convergence curve is far above 1e-16 which is double data precision.

Nevertheless, from what You have written it seems more clear to me, why dt was kept constant in the article - probably the authors wanted to show the performance of RBF interpolation in their advection solver, not affected by accuracy of RK scheme changing with dt.

If You don't mind me asking - I would be very grateful if You found some time to shed some light on the problem stated in the first question about the eigenvalues falling out off RK stability region.

One again thank You for help!