pd1 : Stationary States (1D Problem).

This demo uses Euler's method to locate a stationary solution of a nonlinear parabolic PDE, followed by continuation of this stationary state in a free problem parameter. The equation is

$$\frac{\partial u }{ \partial t} = D~\frac{\partial^2 u }{ \partial x^2} ~+~ p_1~ u ~( 1-u) ,$$

on the space interval $[0,L]$, where $L=$ PAR(11) $=10$ is fixed throughout, as is the diffusion constant $D=$ PAR(15) $=0.1$. The boundary conditions are $u(0) = u(L) = 0$ for all time.

In the first run the continuation parameter is the independent time variable, namely PAR(14), while $p_1=1$ is fixed. The AUTO -constants DS, DSMIN, and DSMAX then control the step size in space-time, here consisting of PAR(14) and $u(x)$. Initial data are $u(x)=\sin(\pi x/L)$ at time zero. Note that in the subroutine STPNT the initial data must be scaled to the unit interval, and that the scaled derivative must also be provided; see the equations-file pv1.f. In the second run the continuation parameter is $p_1$.

Euler time integration is only first order accurate, so that the time step must be sufficiently small to ensure correct results. Indeed, this option has been added only as a convenience, and should generally be used only to locate stationary states.

Table 16.1: Commands for running demo pd1.

AUTO -COMMAND	ACTION
! mkdir pd1	create an empty work directory
cd pd1	change directory
demo('pd1')	copy the demo files to the work directory
run(c='pd1.1')	time integration towards stationary state
sv('1')	save output-files as b.1, s.1, d.1
run(c='pd1.2',s='1')	continuation of stationary states; read restart data from s.1. constants changed : IPS, IRS, ICP, etc.
sv('2')	save output-files as b.2, s.2, d.2