]> https://gitweb.dealii.org/ - dealii-svn.git/commitdiff
Re-generate movie
authorbangerth <bangerth@0785d39b-7218-0410-832d-ea1e28bc413d>
Fri, 3 Nov 2006 22:42:06 +0000 (22:42 +0000)
committerbangerth <bangerth@0785d39b-7218-0410-832d-ea1e28bc413d>
Fri, 3 Nov 2006 22:42:06 +0000 (22:42 +0000)
git-svn-id: https://svn.dealii.org/trunk@14147 0785d39b-7218-0410-832d-ea1e28bc413d

deal.II/examples/step-25/doc/intro.dox
deal.II/examples/step-25/doc/step-25.1d-breather.gif

index 46081e898b896eeba609ab46516ae581bfd0def7..3ccfb2638f28065d3505d6069e97fbe7c3b6aea0 100644 (file)
@@ -161,3 +161,33 @@ auxilliary equation, i.e. $S_j(f,g) = \left( \sin\left[ \theta f +
 nonlinear term in the Jacobian matrix of $F(\cdot)$, i.e. $N_{ij}(f,g)
 = \left( \cos\left[ \theta f + (1-\theta) g\right]\varphi_i, \varphi_j
 \right)_{\Omega}$.
+
+
+<h3>The testcase</h3>
+
+There are a few analytical solutions for the sine-Gordon equation, both in 1d
+and 2d. In particular, the program as is computes the solution to a single
+kink-like solitary wave problem.  This solution is given by Leibbrandt in \e
+Phys. \e Rev. \e Lett. \b 41(7), and is implemented in the
+<code>ExactSolution</code> class.
+
+It should be noted that this closed-form solution strictly speaking only holds
+for the infinite-space initial-value problem (not the Neumann
+initial-boundary-value problem under consideration here). However, given that
+we impose \e zero Neumann boundary conditions, we expect that the solution to
+our initial-boundary-value problem would be close to the solution of the
+infinite-space initial-value problem, if reflections of waves off the
+boundaries of our domain do \e not occur.
+
+The constants $\vartheta$ and $\lambda$ in the 2D solution and $\vartheta$,
+$\phi$ and $\tau$ in the 3D solution are called the B&auml;cklund
+transformation parameters. They control such things as the orientation and
+steepness of the kink. For the purposes of testing the code against the exact
+solution, one should choose the parameters so that the kink is aligned with
+the grid
+
+In 1D, more interesting analytical solutions are known. Many of them are
+listed on http://mathworld.wolfram.com/Sine-GordonEquation.html . We have
+implemented the one kink, two kink, kink-antikink and stationary breather
+solitary-wave solutions.
+
index 9d38a394144ab132cb74a400e26a01b186e3538a..63b098015c773249f39ab4dd865e2d19cf17ca8c 100644 (file)
Binary files a/deal.II/examples/step-25/doc/step-25.1d-breather.gif and b/deal.II/examples/step-25/doc/step-25.1d-breather.gif differ

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