Webpages of the old version pde2path 1.0,
outdated!
We strongly recommend to switch to
pde2path 2.*
Demos
Here we list the examples that have been implemented in the demos directories
of pde2path, together with a few words what makes each example
interesting. See also PDF manual for more details.
Each demo comes with a $\tt xxcmds.m$-file containing a number of
useful commands, and with a $\tt xxdemo.m$-file which guides
through the steps.
Click on the directory names to open the m2html documentation of a demo (in
a new frame), or click
here
to open the full m2html demos directory.
Scalar equations
bratu
Bratu's problem:
$-\Delta u-f(u,\lam)=0, \quad f(u,\lam)=-10(u-\lam \exp(u)), \quad
u=u(x)\in\R$,
on the unit square with Neumann boundary conditions, i.e.,
$x\in \Om=(-1/2,1/2)^2,\quad \pa_n u|_{\pa\Om}=0$.
This is used as our main introductory example to pde2path
in Basics.
ac
A cubi-quintic Allen--Cahn equation on a rectangle with Dirichlet Boundary
conditions:
$$-\mu\Delta u-\lam u-u^3+u^5=0\quad\text{on}\quad \Om=[-L_x,L_x]
\times[-L_y,L_y], \quad u|_{\pa\Om}=0
$$
with two parameters $\mu>0$ and $\lam\in\R$. This in particular serves
as an example for ad-hoc parameter switching and for time integration.
achex
The Allen--Cahn equation on a hexagonal domain with various BC: an example
for (slightly) more complicated domains and boundary conditions.
acgc
Allen-Cahn with global coupling: $G(u,\lam):=
-0.1\Delta u-u-u^3+u^5-\lam\spr{u}=0$ on $\Om=[-\pi/2,\pi/2]^2,
\quad u|_{\pa\Om}=0$, where $\spr{u}=\int_\Om u\mathrm{d} x$.
The global coupling cannot be handled directly with the standard
sparse linear algebra, hence this is an example for simple customization
of pde2path, here by adapting the linear system solvers to
deal with the global coupling using a Sherman--Morrison formula.
Systems
chemotax
A chemotaxis model in the form of a quasilinear
2 component reaction diffusion system:
$\newcommand{\bpm}{\begin{pmatrix}}\newcommand{\epm}{\end{pmatrix}}$
$$
0=G(u):=\bpm -D\Delta u_1+\lam \nabla\cdot(u_1\nabla u_2)-r u_1(1-u_1)
\\-\Delta u_2-\frac{u_1}{1+u_1}+u_2\epm,
$$
where the $\lam\in\R$ is called the chemotaxis coefficient and
$D>0$ and $r\in\R$ are additional parameters. The first example
for the coding of systems.
animal Same system as chemotax but
over an animal-shaped domain: example how to put "arbitrary complicated"
geometries into pde2path .
schnakenberg
A Schnakenberg model with
a rather extensive scanning of the bifurcation diagram:
\begin{align*}
0&=\Delta u_1-u_1+u_1^2u_2,\\
0&=d\Delta u_2 +\lambda-u_1^2u_2.\\
\end{align*}
In fact, using cont
the many bifurcations here lead to multiple undesired
branch switching. This was our first motivation to set up
pmcont.
Download a movie of patterns in
the Schnakenberg model, generated by pde2path.
gpsol
(Systems of) Gross--Pitaevskii equations describing, e.g.,
Bose--Einstein (vector) solitons. Scalar case:
$$
\ri\pa_t\psi=-\Delta\psi+r^2\psi-\sigma|\psi|^2\psi,
$$
where $\psi=\psi(x,y,t)\in\C$, $r^2=x^2+y^2$, and $\sigma=1$
(focussing case). After some transformations
we obtain the 2-component real elliptic system
$$\begin{split}
&-\Delta u+(r^2-\mu)u-|U|^2 u-\om(x\pa_y v-y\pa_x v)=0, \\
&-\Delta v+(r^2-\mu)v-|U|^2 v-\om(y\pa_x u-x\pa_y u)=0,
\end{split}
$$
where $|U|^2=u^2+v^2$.
Similarly we can treat the system case
$$
\ri\pa_t\psi_1=[-\Delta+r^2-\sigma|\psi_1|^2-g_{12}|\psi_2|^2]\psi_1,
\quad \ri\pa_t\psi_2=[-\Delta+r^2-\sigma|\psi_2|^2-g_{21}|\psi_1|^2]\psi_2,
$$
with interspecies interaction coefficients $g_{12},g_{21}$.
This gives an elliptic system of four real equations.
These examples may serve as templates for the coding of nontrivial ${\tt b}$.
Moreover, searching for localized solutions (multipoles, azimuthons and
vortices) these examples show the highly efficient local mesh refinement,
and some tricks for initialization, i.e., finding good starting points.
rbconv
2D Rayleigh-Benard convection
in the streamfunction formulation over a rectangular domain,
$$
\begin{split}
-\Delta\psi + \omega&= 0, \\
-\sigma\Delta \omega -\sigma R \partial_x\theta + \partial_x\psi\partial_z\omega - \partial_z\psi\partial_x\omega &=0, \\
-\Delta \theta - \partial_x\psi + \partial_x\psi\partial_z\theta
- \partial_z\psi\partial_x\theta &=0,
\end{split}
$$
with streamfunction $\psi$, temperature $\theta$, and the auxiliary
$\omega=\Delta \psi$. In particular we discuss no-slip versus stress-free
boundary conditions.
vkplate
The von Karman equations for buckling of an elastic plate $\Om=[-l_x,l_x]\times [-l_y,l_y]\subset\R^2$
are given by
$\huga{
\begin{split}
-\Delta^2 v-\lam\pa_x^2 v+[v,w]&=0, \quad %\\
-\Delta^2 w-\frac 1 2 [v,v]=0,
\end{split}
\label{vk1}
}$
Here $v:\Om\to\R$ is the out of plane deformation, $w:\Om\to\R$ is
the Airy stress function, $\Delta^2=(\pa_x^2+\pa_y^2)^2$ is the
squared Laplacian, $\lam$ is the compression parameter, and the
bilinear form $[\cdot,\cdot]$ is given by
$$
[v,w]:=v_{xx}w_{yy}-2v_{xy}w_{xy}+v_{yy}w_{xx}.
$$
There are a number of choices for the boundary conditions for
$\reff{vk1}$. We consider partially clamped plates
which show some interesting secondary bifurcations also called
mode jumping.
Clearly, $\reff{vk1}$, containing the squared Laplacian,
must first be rewritten as an elliptic system suitable for pde2path.
Our choice is to treat $\reff{vk1}$ as a 10 component
system (including a slight regularization). This serves as an example
for coding some high--dimensional system and nontrivial boundary conditions.