brf : Finite Differences in Space.

This demo illustrates the computation of stationary solutions and periodic solutions to systems of parabolic PDEs in one space variable. A fourth order accurate finite difference approximation is used to approximate the second order space derivatives. This reduces the PDE to an autonomous ODE of fixed dimension which AUTO is capable of treating. The spatial mesh is uniform; the number of mesh intervals, as well as the number of equations in the PDE system, can be set by the user in the file brf.inc.

As an illustrative application we consider the Brusselator (HoKnKu:87 HoKnKu:87)

$$\begin{array}{cl} u_t &= {D_x / L^2} u_{xx} + u^2v - (B+1)u + A, \\ v_t &= {D_y / L^2} v_{xx} - u^2v + Bu, \\ \end{array}$$ (16.4)

with boundary conditions $u(0,t)=u(1,t)=A$ and $v(0,t)=v(1,t)=B/A$.

Note that, given the non-adaptive spatial discretization, the computational procedure here is not appropriate for PDEs with solutions that rapidly vary in space, and care must be taken to recognize spurious solutions and bifurcations.

Table 16.5: Commands for running demo brf.

AUTO -COMMAND	ACTION
! mkdir brf	create an empty work directory
cd brf	change directory
demo('brf')	copy the demo files to the work directory
run(c='brf.1')	compute the stationary solution family with Hopf bifurcations
sv('brf')	save output-files as b.brf, s.brf, d.brf
run(c='brf.2',s='brf')	compute a family of periodic solutions from the first Hopf point. Constants changed : IRS, IPS
ap('brf')	append the output-files to b.brf, s.brf, d.brf
run(c='brf.3',s='brf')	compute a solution family from a secondary periodic bifurcation. Constants changed : IRS, ISW
ap('brf')	append the output-files to b.brf, s.brf, d.brf