pd2 : Stationary States (2D 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 equations are

$$\begin{array}{cl} {\partial u_1 / \partial t} &= D_1~{\partia... ...tial^2 u_2 / \partial x^2} ~-~ u_2 ~+~ u_1 u_2 , \\ \end{array}$$ (16.1)

on the space interval $[0,L]$, where $L=$ PAR(11) $=1$ is fixed throughout, as are the diffusion constants $D_1=$ PAR(15) $=1$ and $D_2=$ PAR(16) $=1$. The boundary conditions are $u_1(0) = u_1(L) = 0$ and $u_2(0) = u_2(L) = 1$, for all time.

In the first run the continuation parameter is the independent time variable, namely PAR(14), while $p_1=12$ is fixed. The AUTO -constants DS, DSMIN, and DSMAX then control the step size in space-time, here consisting of PAR(14) and $(u_1(x),u_2(x))$. Initial data at time zero are $u_1(x)=\sin(\pi x/L)$ and $u_2(x)=1$. Note that in the subroutine STPNT the initial data must be scaled to the unit interval, and that the scaled derivatives must also be provided; see the equations-file pv2.f. In the second run the continuation parameter is $p_1$. A branch point is located during this run.

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 16.2: Commands for running demo pd2.

AUTO -COMMAND	ACTION
! mkdir pd2	create an empty work directory
cd pd2	change directory
demo('pd2')	copy the demo files to the work directory
run(c='pd2.1')	time integration towards stationary state
sv('1')	save output-files as b.1, s.1, d.1
run(c='pd2.2',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