Solving one-dimensional PDE's using the PDE Toolbox

Hi,

I've been trying to solve a non-linear, heat-equation-type system of PDE's using the 'pdepe' function, with only one dimension in space. However, for many sets of parameter values, the solver exhibits unstable behaviour (oscillations, etc). I think that my problem demands a more sophisticated solver, due to nonlinearities and discontinuities. Hence, I am now trying to use the PDE Toolbox, hoping that it would be able to handle my problem, since it has an adaptive mesh algorithm, etc. However, I was unable to figure out how to use the PDE Toolbox for one-dimensional problems, nor was I able to find any examples of this.

Is it possible to use the PDE Toolbox to solve one-dimensional problems? I'm find with using the command-line interface (I don't necessarily need to use the GUI). If this is possible, I'd greatly appreciate it if someone could provide me with some code that solves a simple heat equation PDE (one dimension in space) using the PDE Toolbox, just to get me on the right foot (since, for some reason, I get really lost when I try to read the documentation for the PDE Toolbox).

Thanks,

Abed

Answer 1

[Bill Greene]

Hi,

Solving a 1-D PDE with PDE Toolbox is fairly straightforward. You just define a rectangular region of the appropriate width and arbitrary height. On the two edges at y=constant, you want a zero-Neumann BC. On the edges at x=constant, the BCs should not vary in y.

I've appended a very simple example of time-dependent heat transfer in a bar below.

Based on your description of the problem you are trying to solve, however, I can't think of any reason why PDE Toolbox should give a better solution than pdepe.

Bill

function simple1DTest
% 1D transient heat transfer in x-direction
h =.1; w=1; % width equal one, height is arbitrary (set to .1)
g = decsg([3 4 0 w w 0 0 0 h h]', 'R1', ('R1')');
[p, e, t]=initmesh(g, 'Hmax', .05);
b=@boundFile;
c=1; a=0; d=1;
f='100*x'; % heat load varies along the bar
u0 = 0; % initial temperature equals zero
tlist = 0:.02:1;
u=parabolic(u0, tlist, b,p,e,t,c,a,f,d);
figure; pdeplot(p,e,t, 'xydata', u(:,end), 'contour', 'on'); axis equal;
title(sprintf('Temperature Distribution at time=%g seconds', tlist(end)));
figure; plot(tlist, u(2,:)); xlabel 'Time'; ylabel 'Temperature'; grid;
title 'Temperature at tip as a function of time'
end

function [ q, g, h, r ] = boundFile( p, e, u, time )
N = 1; ne = size(e,2);
q = zeros(N^2, ne); g = zeros(N, ne);
h = zeros(N^2, 2*ne); r = zeros(N, 2*ne);
% zero Neumann BCs (insulated) on edges 1 and 3 at y=0 and y=h
% and the right edge, edge 2
for i=1:ne
  switch(e(5,i))
    case 4
      h(1,i) = 1; h(1,i+ne) = 1;
      r(i) = 500; r(i+ne) = 500; % 500 on left edge
  end
end
end

[Abed Alnaif]

Thanks, I'll give it a shot.

Answer 2

[tom_brot]

Ok this question has already been answered, but I want to share my experience with pdepe.

I also had problems with oscilating, unstable behaviour of pdepe. I found that the Problem lies not directly in the pdepe, but in the underlying ode15s solver. You can specify some options (with odeset()) for the pdepe-solver which are then passed to the ode15s solver.

The problem is, that only very few of the possible odeset() options are passed. To get a much more stable solution from the underlying ode15s solver I reduced the 'MaxOrder' option from the Default value of {5} to 2.

It's not that hard to make the option available for your pdepe solver.

How to make your pdepe solver more stable:

1) Find the file 'pdepe.m' in somewhere in your matlab installation path and copy it to your working directory. And rename it to e.g. 'pdepe_opt.m'.

2) Find the file 'pdentrp.m' (which is located in the 'private' directory next to 'pdepe.m') and copy it to your working directory.

3) In your local copy of pdepe.m, namely pdepe_opt.m add the line

maxorder = odeget(options,'MaxOrder',[],'fast');

after the line

maxstep = odeget(options,'MaxStep',[],'fast');

4) Then add

'MaxOrder',maxorder,

as argument of the odeset command, which is some lines below.

5) Before the line in your code where you call pdepe, insert

options = odeset('MaxOrder',2)

(or any number from 1 to 5) and then call the optimized pdepe function with your specified options:

sol = pdepe_opt(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t,options)

I hope this works as well for you as it did for me.

Best regards,

Tom

[Abed Alnaif]

Thanks, that's a neat solution!

[Sonia]

It worked for me too, after trying out all other impossible options! In fact, I didn´t need to modify the function, passing extra option in odeset sufficed. Thanks a lot!!

[Lucy]

This is amazing thank you!

[Mario Buchely]

This solution didn't work for me.

But I modified the pdepe.m fuction, and changed the solver from ode15s to ode23s. After this change, and copying the modifed in the workind directory (and the pdentrp.m file), my solution converged and worked fine.

Thank you for the key to solve this issue.