From 4ea3a176793c0be68b91520e997e262a5cdf9bc2 Mon Sep 17 00:00:00 2001 From: Martin Kronbichler Date: Fri, 27 Mar 2020 14:14:44 +0100 Subject: [PATCH] Further improve text --- doc/doxygen/references.bib | 13 +++++++++++++ examples/step-67/doc/intro.dox | 4 ++-- examples/step-67/doc/results.dox | 16 +++++++++------- 3 files changed, 24 insertions(+), 9 deletions(-) diff --git a/doc/doxygen/references.bib b/doc/doxygen/references.bib index ad48c5484f..2073a9fb2d 100644 --- 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 02a599009e..7fc0277f54 100644 --- 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 effective 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 a073075189..321f9bb43d 100644 --- 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. 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. -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: @@ -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. Gassner (2013) 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 -- 2.39.5