1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
|
clear all; clf
% parameters
N=100; % number of gridpoints
L=10; % domain length
mu=0.01; %
dt=0.05; % time step
x0=L/2; %x-coordinate at initial time
l0=0.5; length width
% differentiation matrices
scale=-2/L;
[x,DM] = chebdif(N,2);
dx=DM(:,:,1)*scale;
dxx=DM(:,:,2)*scale^2;
x=(x-1)/scale;
Z=zeros(N,N); I=eye(N);
% system matrices
E=I;
A=mu*dxx;
% boundary conditions
loc=[1;N];
E(loc,:)=0;
A(loc,:)=[I(1,:)-I(N,:); dx(1,:)-dx(N,:)];
% march in time matrix
Mm=(E-A*dt/2);
M=Mm\(E+A*dt/2);
% initial condition
%q=1*0+0.1*[x*0+exp(-((x-x0)/l0).^2)];
q=0.1*sin(2*pi*x/L);
% marching loop
quit=0;
while ~quit
nl=0.1*q.*(1-q);
nl(loc)=0;
qnext=M*q+Mm\nl*dt; % one step forward
% stop when nothing moves
norm(qnext-q,2)<0.01
q=qnext;
% plotting
%subplot(1,2,1);
plot(x,q,'b');
axis([0,L,-2,2])
drawnow
end
legend('position'); title('Vibrating string')
xlabel('x'); ylabel('f');
|
