From 060255b27c1da21dd92469ed6d1a71c1c0cf271c Mon Sep 17 00:00:00 2001 From: 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 ++++++++++++++++++ .../step-25/doc/step-25.1d-breather.gif | Bin 397752 -> 69649 bytes 2 files 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. + diff --git a/deal.II/examples/step-25/doc/step-25.1d-breather.gif b/deal.II/examples/step-25/doc/step-25.1d-breather.gif index 9d38a394144ab132cb74a400e26a01b186e3538a..63b098015c773249f39ab4dd865e2d19cf17ca8c 100644 GIT binary patch literal 69649 zcmeFaWmH>H+pbFs72GKWin|ls-K_+QmE!L1R%mez?k>UI-Q5et-Q7K$^p)@1=lnT; z&e`MLdv}ct5?EPttt5<#d*07IA0(wExVZI%UY)_*z`(q`yuiS~WM*dmx$A&|@qOul zd4V}O=@5c`fgvXDcClDg=z!7n?a=jw@r4n3@nzQa_0@$D zf^NRhA)t@Ze06nwp?m)f_X48}vsuw0)Zwd}*`W(Pt|Jq=1A3$`G4xrw#L!Jhh*${v zbYk5VWCUBqq)ze({A4OicXe zY(l=ix_?IXg$~{UJw)gw6MEBq$+Ufej{1)gU!d>G|34eaJ={MACV4?r5LZza75*#^ zU}T1U1^prYpB;ZbCYW~+Ul>3CA3p;EgMvds!@?sXqoP4Ev2pPUiAl*RscGpMnOWI6 zxq0~oh2Wy%lG3vBipr|$n%cVhhQ_Amme#iRj?S*`p5DIxfx)5Skl)nCVtHh zO!>{vEiHF0uB>m=OwR;u@9iHP9vz>Yo}FJ@UR~eZ-re7nto__+e}VDFZzv+l@}2xa zvQ=xV*8O=A3B^4zRA1chwNjj{;kl}0B(9g23cG!(v}K)4x)!(GHcbM(0cFV><$QT9 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