Computer Algebra Recipes

6.3. NUMERICAL SIMULATION OF PDES 277
evaluates the time step size k = rh 2 ,aswellastheparameterc. Wanting to
animate his numerical solution, he decides to create N = 50 plots, with each
plotseparatedintimebys =20steps.
> r:=0.05: h:=1.0/(M+1); k:=r*h^2; c:=1-2*r; N:=50: s:=20:
h := 0:08333333333 k := 0:0003472222222 c := 0:90
An operator f will be used to input the initial parabolic temperature pro¯le.
> f:=x->evalf(100*x*(1-x)):
With the help of f, the initial (j = 0 time row) temperatures at the M internal
mesh points are calculated, and expressed as a column vector4 T0 so that matrix
multiplication can be performed.
> T[0]:=; #input temperatures
2
6
T0 := 6
4
7:638888889
13:88888889
18:75000000
22:22222222
24:30555556
25:00000000
24:30555556
22:22222222
18:75000000
13:88888889
7:638888886
A graphing operator is formed to plot the temperature pro¯le on the jth step,
the plotting points being connected by straight lines. The zero temperatures at
the endpoints are included.
> gr:=j->plot([[0,0],seq([i/(M+1),T[j][i,1]],i=1..M),[1,0]],
labels=["x","T"]):
In the following do loop, the temperature pro¯le is calculated every s=20steps. > for n from 1 to N do
> T[n]:=A^s . T[n-1];
> end do:
The CPU time on a 3-GHz personal computer to perform the calculation
> CPUtime:=(time()-begin)*seconds;
CPUtime := 0:350 seconds
is a fraction of a second, hardly worth recording. The plots[display] command
5 with the insequence=true option is employed to produce an animated
sequence of the temperature pro¯le in the rod.
> plots[display]([seq(gr(j),j=0..N)],insequence=true);
4 Entered with the short-hand notation >.
5 Ashortcutto¯rstenteringwith(plots) and then display.
3
7
5