infrastructure code supporting AMR.
Supporting this point is the fact that although there are
- <a href="https://www.dealii.org/publications.html">more than 1,000
+ <a href="https://www.dealii.org/publications.html">more than 2,000
publications</a> presenting results obtained with deal.II, we are aware of
only a relatively small number of applications that have been built with deal.II from
scratch; all others have started as modifications of one of the tutorial
url = {http://doi.org/10.1016/j.camwa.2016.09.004},
}
+% ------------------------------------
+% Step 59
+% ------------------------------------
+@article{Lynch1964,
+author = {Lynch, Robert E. and Rice, John R. and Thomas, Donald H.},
+title = {Direct Solution of Partial Difference Equations by Tensor Product Methods},
+year = {1964},
+issue_date = {December 1964},
+publisher = {Springer-Verlag},
+address = {Berlin, Heidelberg},
+volume = {6},
+number = {1},
+issn = {0029-599X},
+url = {https://doi.org/10.1007/BF01386067},
+doi = {10.1007/BF01386067},
+journal = {Numer. Math.},
+month = {dec},
+pages = {185–199},
+numpages = {15}
+}
% ------------------------------------
% Step 61
% ------------------------------------
narrow case, we refrain from pursuing this idea.
Interestingly, the exact inverse of the matrix $L$ can be found through tensor
-products due to a method introduced by <a
-href="http://dl.acm.org/citation.cfm?id=2716130">R. E. Lynch, J. R. Rice,
-D. H. Thomas, Direct solution of partial difference equations by tensor
-product methods, Numerische Mathematik 6, 185-199</a> from 1964,
+products due to a method introduced by Lynch et al. @cite Lynch1964 from 1964,
@f{align*}{
L^{-1} &= S_1 \otimes S_0 (\Lambda_1 \otimes I + I \otimes \Lambda_0)^{-1}
S_1^\mathrm T \otimes S_0^\mathrm T,
}
- // The following function does an entire stage of a Runge--Kutta update and is
- // - alongside the slightly modified setup - the heart of this tutorial
+ // The following function does an entire stage of a Runge--Kutta update
+ // and is, alongside the slightly modified setup, the heart of this tutorial
// compared to step-67.
//
// In contrast to step-67, we are not executing the advection step
These pictures show that what we find here is in accordance with what one
-typically sees in other publications on the topic @cite Bendse2004. Maybe more interestingly, the
+typically sees in other publications on the topic @cite Bendse2004.
+Maybe more interestingly, the
result looks like a truss bridge (except that we apply the load at the top of
the trusses, rather than the bottom as in real truss bridges, akin to a "deck
-truss" bridge), suggesting that the designs that have been used in bridge-
-building for centuries are indeed based on ideas we can now show to be optimal
+truss" bridge), suggesting that the designs that have been used in bridge-building
+for centuries are indeed based on ideas we can now show to be optimal
in some sense.
-
-
<h4>Possibilities for extensions</h4>
The results shown above took around 75 iterations to find, which is quite