Allen-Cahn
This demo shows how AUTO can be used for the time-integration
of a nonlinear parabolic PDE, followed by the continuation of
a stationary state found in the time integration.
The equations have the form of an Allen-Cahn equation, namely,
u_t = c u_xx + f(u,lambda) ,
where u = u(x,t) satisfies the boundary conditions
u(0,t) = u(L,t) = 0
and has initial value u(x,0) = g(x) at time t=0.
To run this demo, type "auto ace.auto". To plot the solutions
during the time evolution, type "@pp ivp". To see the integral
of the solutions as a function of time, click on "bifurcation",
and select "time" as the "X"-axis.
One can use plaut04 to see an animation of the solutions: type
"@pl ivp" to see all solutions. Then select "NONE" under Label",
followed by selecting "Highlight Orbit" under "Options".
To plot solutions of the subsequent continuation of the stationary
solution found by time-integration, type "@pp ace". To plot the
corresponding bifurcation diagram click on "bifurcation" in the
pyplaut window. To see the labels of the solutions that have been
saved, type "d2" in the shell window. (The default option "d2" is
set in the file "autorc".) To see the corresponding solutions,
click on "solution".
The time-integration uses the setting IPS=16, and the continuation
of he stationary solution uses IPS=17, as set in the script file
"ace.auto". (See the AUTO manual for the settings of IPS, which
defines the problem type.)
Note that for these values of IPS, only the function f needs to
be defined in the subroutine FUNC in equations-file ace.f90. In
the current demo f is given by
f(u,lambda) = lambda u + u^3 - u^5
The diffusion term "c u_xx" is automatically added by AUTO, with
the diffusion constant c set in STPNT, where its current value
(c=0.25) is assigned to PAR(15).
The variable x is scaled from [0,L] to the unit interval [0,1],
which explains why the boundary conditions in BCND are at x=0 and
x=1. Currently L has value L=2, as assigned to PAR(11) in STPNT.
The initial function g(x) in u(x,0)=g(x) is also specified on the
scaled interval [0,1] in STPNT, where it is set to
g(x) = amp sin(pi x)
with the "amplitude" amp set to amp=0.1. The derivative of g(x)
with respect to x must also be provided, and include the scaling
factor L.
Warning: As is known from the theory of stiff differential
equationis, a large time-step may stabilize unstable stationary
states. Thus taking a large time step (controlled by the AUTO-
constants, DS, DSMIN, and DSMAX) may lead to stationary states
that are actually unstable. (This can have advantages too!)
Reference: Samuel M. Allen and John W. Cahn, "Ground State
Structures in Ordered Binary Alloys with Second Neighbor
Interactions," Acta Met. 20, 423 (1972).