Dear all,
I am writing to seek some suggestions about constructing a
summation-by-part derivative operator [1][2] (i.e. collocation spectral
discontinuous Galerkin) under dealii framework. I started from Step-12
where DG scheme is applied to solve linear steady-state advection. Given
that I am interested in solving some conservation laws with explicit and
implicit schemes later, I think that it is a good start.
Essentially, what I want is to compute a one-dimensional derivative w.r.t
to reference domain (i.e. [image: D_{i,j}=\frac{d l_j}{d\xi}(\hat{\xi}_i)]
where
[image: \hat{\xi}_i] is [image: i]-th Gauss-Lobatto point defined in
[0,1]) and compute contravariant basis vectors (i.e [image:
\vec{a}^{1}=\frac{d\xi}{d\vec{x}}] and [image:
\vec{a}^{2}=\frac{d\eta}{d\vec{x}}]).
According to flag
<https://www.dealii.org/current/doxygen/deal.II/fe__update__flags_8h.html#aa94b67d2fdcc390690c523f28019e52fa778d7108798b47d33186dcedd1cff374>,
we can compute contravariant basis vectors by inverting the matrix given by
"jacobian(const unsigned int quadrature_point)".
However, I am not quite sure how to construct [image: D_{i,j}] under the
framework of step-12. A brutal way is to create dummy objects of
1-d triangulation (i.e. [0,1]), FEValue object, and DoFHandler to
construct it within the cell_worker. However, it is resource-consuming. In
addition, it is not clear to me how to convert 1-D indices into the actual
indices. For example for degree=1, we have:
(0,1) - - - (1,1)
| |
| |
(0,0) - - - (0,1)
for Lagrange polynomial [image: \phi_{i,j}=l_i(\xi)l_j(\eta)]. Does the
pair (i,j) enumerate in the following way?
(2) - - - (3)
| |
| |
(0) - - - (1)
On top of that, are quadrature points enumerated in a similar fashion?
Any suggestions or comments are greatly welcome.
Best Regards,
Jau-Uei Chen
References:
[1] Gassner, Gregor J. "A skew-symmetric discontinuous Galerkin spectral
element discretization and its relation to SBP-SAT finite difference
methods." SIAM Journal on Scientific Computing 35.3 (2013): A1233-A1253.
[2] Gassner, Gregor J., Andrew R. Winters, and David A. Kopriva. "Split
form nodal discontinuous Galerkin schemes with summation-by-parts property
for the compressible Euler equations." Journal of Computational Physics 327
(2016): 39-66.
--
The deal.II project is located at http://www.dealii.org/
For mailing list/forum options, see
https://groups.google.com/d/forum/dealii?hl=en
---
You received this message because you are subscribed to the Google Groups
"deal.II User Group" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To view this discussion on the web visit
https://groups.google.com/d/msgid/dealii/bb0e341d-b1ad-48f2-9eaa-91e9f8411e22n%40googlegroups.com.
