Further improve text

diff --git a/doc/doxygen/references.bib b/doc/doxygen/references.bib
index ad48c5484fb4d07ca2e83c43659ed00dac580e95..2073a9fb2d7bbf691cde702877de2a64a10a01cd 100644 (file)
--- a/doc/doxygen/references.bib
+++ b/doc/doxygen/references.bib
@@ -532,6 +532,19 @@ doi = {10.1145/3325864},
 url = {https://doi.org/10.1145/3325864}
 }
 
+@article{Gassner2013,
+  doi = {10.1137/120890144},
+  url = {https://doi.org/10.1137/120890144},
+  year = {2013},
+  volume = {35},
+  number = {3},
+  pages = {A1233--A1253},
+  author = {Gregor J. Gassner},
+  title = {A Skew-Symmetric Discontinuous {G}alerkin Spectral Element Discretization and Its Relation to {SBP}-{SAT} Finite Difference Methods},
+  journal = {{SIAM} Journal on Scientific Computing}
+}
+
+
 % ------------------------------------
 % Step 69
 % ------------------------------------

diff --git a/examples/step-67/doc/intro.dox b/examples/step-67/doc/intro.dox
index 02a599009e35d17618196b64392cbf9a54b8631d..7fc0277f541ed78dbce4b8fcd869b1af49fb83b8 100644 (file)
--- a/examples/step-67/doc/intro.dox
+++ b/examples/step-67/doc/intro.dox
@@ -388,8 +388,8 @@ dimensionless number $\mathrm{Cr}$ denotes the Courant number and can be
 chosen up to a maximally stable number $\mathrm{Cr}_\text{max}$, whose value
 depends on the selected time stepping method and its stability properties. The
 power $p^{1.5}$ used for the polynomial scaling is heuristic and represents
-the closest fit for polynomial degrees between 1 and 8, see e.g.  @cite
-SchoederKormann2018. In the limit of higher degrees, $p>10$, a scaling of
+the closest fit for polynomial degrees between 1 and 8, see e.g.
+@cite SchoederKormann2018. In the limit of higher degrees, $p>10$, a scaling of
 $p^2$ is more accurate, related to the inverse estimates typically used for
 interior penalty methods. Regarding the <i>effective</i> mesh sizes $h_u$ and
 $h_c$ used in the formula, we note that the convective transport is

diff --git a/examples/step-67/doc/results.dox b/examples/step-67/doc/results.dox
index a073075189c1253d9daa6884b3e4ad6f36fe3cc4..321f9bb43d976c5c12d3d9068aa7bb0dd7b1d004 100644 (file)
--- a/examples/step-67/doc/results.dox
+++ b/examples/step-67/doc/results.dox
@@ -411,7 +411,9 @@ no-penetration condition on the cylinder walls, the flow that
 initially impinges head-on onto to cylinder has to rearrange,
 which creates a big sound wave. The following pictures show the pressure at
 times 0.1, 0.25, 0.5, and 1.0 (top left to bottom right) for the 2D case with
-5 levels of global refinement. We clearly see the discontinuity that
+5 levels of global refinement, using 102,400 cells with polynomial degree of
+5 and 14.7 million degrees of freedom over all 4 solution variables.
+We clearly see the discontinuity that
 propagates slowly in the upstream direction and more quickly in downstream
 direction in the first snapshot at time 0.1. At time 0.25, the sound wave has
 reached the top and bottom walls and reflected back to the interior. From the
@@ -441,7 +443,8 @@ many sound waves all over the place.
 </table>
 
 The next picture shows an elevation plot of the pressure at time 1.0 looking
-from the channel inlet towards the outlet -- here, we can see the large number
+from the channel inlet towards the outlet at the same resolution -- here,
+we can see the large number
 of reflections. In the figure, two types of waves are visible. The
 larger-amplitude waves correspond to various reflections that happened as the
 initial discontinuity hit the walls, whereas the small-amplitude waves of
@@ -453,8 +456,8 @@ the result of the transport accuracy of the high-order DG method.
 
 <img src="https://www.dealii.org/images/steps/developer/step-67.pressure_elevated.jpg" alt="" width="40%">
 
-With 2 levels of global refinement, the mesh and its partitioning on 40 MPI
-processes looks as follows:
+With 2 levels of global refinement with 1,600 cells, the mesh and its
+partitioning on 40 MPI processes looks as follows:
 
 <img src="https://www.dealii.org/images/steps/developer/step-67.grid-owner.png" alt="" width="70%">
 
@@ -708,9 +711,8 @@ error can give rise to an increase in the numerical energy and eventually
 render the discretization unstable. This is because of the inexact numerical
 integration of the terms in the Euler equations, which both contain rational
 nonlinearities and higher-degree content from curved cells. A way out of this
-dilemma are so-called skew-symmetric formulations, see e.g. <a
-href=https://doi.org/10.1137%2F120890144>Gassner (2013)</a> for a simple
-variant. Skew symmetry means that switching the role of the solution
+dilemma are so-called skew-symmetric formulations, see @cite Gassner2013 for a
+simple variant. Skew symmetry means that switching the role of the solution
 $\mathbf{w}$ and test functions $\mathbf{v}$ in the weak form produces the
 exact negative of the original quantity, apart from some boundary terms. In
 the discrete setting, the challenge is to keep this skew symmetry also when