[AdrianN]

Dear all,

I have implemented a DG-ADER scheme for the resolution of linear systems of hyperbolic conservation laws, eg. the linearized Euler equations for aeroacustics. I am trying to figure out why the scheme becomes oscillatory for a gaussian initial condition when using quadrilateral unstructured grid while it is completely non-oscillatory when using structured meshes where the cell have a random quadrilateral shape.

Please find attached three snapshots of three different solutions after several time steps: a) using a regular square grid where two columns have different width, b) using a structured trapezoidal grid and c) using an unstructured grid. As mentioned before, it can be observed that oscillations appear for case c).

The CFL has been reduced to obtain a time step much lower than the bounding value, which depends on the cell size, dimensions of the problem and degree of the polynomials. The degree of the polynomials is 2.

If someone could give me any explanation for this, it would be very helpful.

[FMDenaro]

why do you plot the constant-cell value instead of isolines?

however, could you plot the initial condition and the solution after a single time step on the unstructured grid? I suppose you have both pressure and velocity as well as you can superimpose also the analytical solution

[AdrianN]

Thanks for your quick answer. I plot cell averaged values because I still have to figure out how to represent discontinouous arbitrary polynomial surfaces inside each cell with Paraview.

Here you can find the results for the unstructured grid after only one time step, for both the pressure and the velocity in x. I do not include the velocity in y as it exhibits the same behavior. You can also find the plots of the reference solution, represented with the color scale range of the numerical solution.

Those spurious oscillations in the velocity do not appear when using any kind of structured mesh but always appear for unstructured meshes like this one.

I don't know if I have any bug in the code that is generating such oscillations, but I believe that if so, even in the structured mesh with trapezoidal elements the problem would be noticeable, as the jacobians for the coordinate transformations are not trivial.


Kind regards,

[FMDenaro]

What is strange is that the velocity field has no negative values...in an oscillating solution, the first appearence is the production of spurious minimum...
however the pressure field appears with spurious modes...

[FMDenaro]

Anyway, you need to plot exactly the values in the nodes as they are produced by the code. The constant cell-averaged value can produce a wrong analysis. For this reason I suggested to plot the initial condition to exclude that graphics produces a strange pattern also a t=0

[AdrianN]

Thanks for the comments. I will try to figure out how to plot all data provided by the DG solution to see if this helps. Anyway, I have checked that the solution at t=0, represented as cell averages, matches the exact solution.

[FMDenaro]

I suppose that the code does not distinguish between stractured or unstructured data, isn't it? Therefore also the structured grid is treated as unstructured?
Again, it is strange such a pressure pattern after a single time-step...

[AdrianN]

Sorry for the delay, I have been out. As you mention, it does not distinguish between structured and unstructured grid, it treats both cases identically and there are no switching options for one and other.

Could it be caused by the fact of not using any limiting techniques in the DG reconstruction, as it is usually done for the majority of hyperbolic problems? I have not implemented such techniques as the problems I am interested in consist of smooth solution without lost of regularity.

[FMDenaro]

Originally Posted by AdrianN

Sorry for the delay, I have been out. As you mention, it does not distinguish between structured and unstructured grid, it treats both cases identically and there are no switching options for one and other.

Could it be caused by the fact of not using any limiting techniques in the DG reconstruction, as it is usually done for the majority of hyperbolic problems? I have not implemented such techniques as the problems I am interested in consist of smooth solution without lost of regularity.

A non-monotone scheme generates wiggles but that happens on both structured and unstructured grids...As I wrote, the velocity plot should show spurious minima in the solution...

[AdrianN]

Thanks for your help, i will have go through the code in detail again to look for errors...