Matrix-free: two-point volume fluxes for entropy-stable DG?

[Jack Coughlin]

I am interested in using the deal.ii matrix-free infrastructure to implement an entropy-stable DG method. The entropy stable family of methods uses an interesting variation, where the volume flux is actually a consistent two-point flux function (similar to the numerical flux).

I've attached a screenshot of the formula in question, which is taken from this documentation page. I've also attached a review of the method.

As far as I can tell, to do this you would need to be able to tell a class like FEEvaluation to integrate a given submission only for one test function. I think I can see how to do this integration manually with FEValues, but it would be nice to take advantage of the matrix-free infrastructure if possible.

Thank you,

-Jack C

[Martin Kronbichler]

Dear Jack,

Two-point fluxes are something we have not yet included in deal.II. I would be very interested if we could realize this feature. I must admit that I did not yet try to understand the complete formula in the link you provided, but in fact it should be possible to implement this with the internal evaluators used by FEEvaluation very efficiently. Would you be willing to share the FEValues variant, it is enough to provide it in 1D. I did work with the Trixi code before, but did not yet look into the algorithms, so I would be very curious to learn the details. My initial approach would be to encode the full D_split matrix to be fed into the tensor product evaluators in the internal namespace, and then add the respective boundary fluxes. Once we have a basic version, we can think about making the interface nicer.

Best,
Martin

Am 23.04.24 um 17:19 schrieb Jack Coughlin:

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[rru]

Dear Martin,

The conversations for  the two point flux entropy stable DG seem to be deleted. However this is an important topic (for strong shocks etc). Could this be done with the Matrix-Free / FEEvaluation routines or would one need to do something similar to step-69. I prefer FE_DGQ with Gauss-Lobatto quadrature and the highly efficient routines therein.  Naively speaking, we would need the following type of pseudo code (for 1D) :

Form split_derivative_matrix = D_split[q,q1] for q,q1 in [0....N];

(diagonal entries of D_split are zero)

for q in [0:N]: // q and basis same for Gauss-Lobatto quadrature

       get u[q] // dof value

        for q1 in [q+1:N] // Diagonal contribution is 0

             get u1[q]

             F[u,u1] = two_point_flux(u,u1) // This will be symmetric in u,u1 

             u_q += D_split[q,q1]*F[u,u1]

             u_q += D_split[q1,q]*F[u1,u]  // Symmetric so F[u,u1] = F[u1,u]

Maybe a more sophisticated algorithm with Tensor / Hadamard products could be formulated ?

For high performance, can the above be done efficiently in the deal.ii matrix free framework ? And if so how ?

Regards,

Rochan