From: bangerth Date: Sun, 5 Nov 2006 23:22:01 +0000 (+0000) Subject: A bit more X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=a856f3b21966bcb6bc2e206193c6963d81b82605;p=dealii-svn.git A bit more git-svn-id: https://svn.dealii.org/trunk@14152 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-25/doc/results.dox b/deal.II/examples/step-25/doc/results.dox index f31f1fd9bf..7b011cb349 100644 --- a/deal.II/examples/step-25/doc/results.dox +++ b/deal.II/examples/step-25/doc/results.dox @@ -35,13 +35,45 @@ To this end, we can rotate the kink solution discussed above about the $z$ axis, \image html step-25.2d-angled_kink.gif "Animation of a moving 2D kink, at 45 degrees to the axis of the grid, showing boundary effects." width=5cm -Another interesting solution to the sine-Gordon equation (which cannot be obtained analytically) can be produced by using two 1D breathers to construct the following separable 2D initial condition: +Another interesting solution to the sine-Gordon equation (which cannot be +obtained analytically) can be produced by using two 1D breathers to construct +the following separable 2D initial condition: \f[ -u_0(x) = u_{\mathrm{pseudobreather}}(x,t_0) = 16\arctan \left(\frac{m}{\sqrt{1-m^2}} \frac{\sin\left(\sqrt{1-m^2}t_0\right)}{\cosh(mx_1)} \right)\arctan \left(\frac{m}{\sqrt{1-m^2}} \frac{\sin\left(\sqrt{1-m^2}t_0\right)}{\cosh(mx_2)} \right), + u_0(x) = + u_{\mathrm{pseudobreather}}(x,t_0) = + 16\arctan \left( + \frac{m}{\sqrt{1-m^2}} + \frac{\sin\left(\sqrt{1-m^2}t_0\right)}{\cosh(mx_1)} \right) + \arctan \left( + \frac{m}{\sqrt{1-m^2}} + \frac{\sin\left(\sqrt{1-m^2}t_0\right)}{\cosh(mx_2)} \right), \f] -where $x=(x_1,x_2)\in{R}^2$, $m=0.5(<1)$ as in the 1D case we discussed above. For the simulation shown below, we have chosen $\theta=\frac{1}{2}$, $k=10h$, $t_0=-5.4414$ and $t_f=2.7207$. The solution is pretty interesting --- it acts like a breather (as far as the pictures are concerned); however, it appears to break up and reassemble, rather than just oscillate. Naturally, the are things in the animation that can not be explained in a few words due to the fact that this initial condition is not an analytical solution to the sine-Gordon equation. +where $x=(x_1,x_2)\in{R}^2$, $m=0.5(<1)$ as in the 1D case we discussed +above. For the simulation shown below, we have chosen $\theta=\frac{1}{2}$, +$k=10h$, $t_0=-5.4414$ and $t_f=2.7207$. The solution is pretty interesting +--- it acts like a breather (as far as the pictures are concerned); however, +it appears to break up and reassemble, rather than just oscillate. Naturally, +the are things in the animation that can not be explained in a few words due +to the fact that this initial condition is not an analytical solution to the +sine-Gordon equation. \image html step-25.2d-pseudobreather.gif "Animation of a 2D pseudobreather." width=5cm

Future work

Time-adaptive grids would likely be useful when solving a problem like this one (especially in the case of a traveling wave). However, due to the complexity of their implementation they are better left for another day. + + + + +

Possibilities for extensions

+ +It is instructive to change the initial conditions. Most choices will not lead +to solutions that stay localized, but lead to solutions where the wave-like +character of the equation dominates and a wave travels away from the location +of a localized initial condition. For example, it is worth playing around with +the InitialValues class, by replacing the call to the +ExactSolution class by something like this function: +@f[ + u_0(x,y) = \cos\left(\frac x2\right)\cos\left(\frac y2\right) +@f] +if $|x|,|y|\le \frac\pi 2$, and $u_0(x,y)=0$ outside this region.