From 6fbacd57dcb7d610d17a97336dbf034aabe1828d Mon Sep 17 00:00:00 2001 From: Wolfgang Bangerth Date: Fri, 3 Nov 2006 22:42:06 +0000 Subject: [PATCH] Re-generate movie git-svn-id: https://svn.dealii.org/trunk@14147 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-25/doc/intro.dox | 30 ++++++++++++++++++++++++++ 1 file changed, 30 insertions(+) diff --git a/deal.II/examples/step-25/doc/intro.dox b/deal.II/examples/step-25/doc/intro.dox index 46081e898b..3ccfb2638f 100644 --- a/deal.II/examples/step-25/doc/intro.dox +++ b/deal.II/examples/step-25/doc/intro.dox @@ -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. $N_{ij}(f,g) = \left( \cos\left[ \theta f + (1-\theta) g\right]\varphi_i, \varphi_j \right)_{\Omega}$. + + +

The testcase

+ +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 +ExactSolution 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 Bäcklund +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. + -- 2.39.5