Error in FDM for a coupled PDEs - matlab

Question

Here is the code which is trying to solve a coupled PDEs using finite difference method,
clear;
Lmax = 1.0; % Maximum length
Wmax = 1.0; % Maximum wedth
Tmax = 2.; % Maximum time
% Parameters needed to solve the equation
K = 30; % Number of time steps
n = 3; % Number of space steps
m =30; % Number of space steps
M = 2;
N = 1;
Pr = 1;
Re = 1;
Gr = 5;
maxn=20; % The wave-front: intermediate point from which u=0
maxm = 20;
maxk = 20;
dt = Tmax/K;
dx = Lmax/n;
dy = Wmax/m;
%M = a*B1^2*l/(p*U)
b =1/(1+M*dt);
c =dt/(1+M*dt);
d = dt/((1+M*dt)*dy);
%Gr = gB*(T-T1)*l/U^2;
% Initial value of the function u (amplitude of the wave)
for i = 1:n
if i < maxn
u(i,1)=1.;
else
u(i,1)=0.;
end
x(i) =(i-1)*dx;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for j = 1:m
if j < maxm
v(j,1)=1.;
else
v(j,1)=0.;
end
y(j) =(j-1)*dy;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for k = 1:K
if k < maxk
T(k,1)=1.;
else
T(k,1)=0.;
end
z(k) =(k-1)*dt;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Value at the boundary
%for k=0:K
%end
% Implementation of the explicit method
for k=0:K % Time loop
for i=1:n % Space loop
for j=1:m
u(i,j,k+1) = b*u(i,j,k)+c*Gr*T(i,j,k+1)+d*[((u(i,j+1,k)-u(i,j,k))/dy)^(N-1)*((u(i,j+1,k)-u(i,j,k))/dy)]-d*[((u(i,j,k)-u(i,j-1,k))/dy)^(N-1)*((u(i,j,k)-u(i,j-1,k))/dy)]-d*[u(i,j,k)*((u(i,j,k)-u(i-1,j,k))/dx)+v(i,j,k)*((u(i,j+1,k)-u(i,j,k))/dy)];
v(i,j,k+1) = dy*[(u(i-1,j,k+1)-u(i,j,k+1))/dx]+v(i,j-1,k+1);
T(i,j,k+1) = T(i,j,k)+(dt/(Pr*Re))*{(T(i,j+1,k)-2*T(i,j,k)+T(i,j-1,k))/dy^2-Pr*Re{u(i,j,k)*((T(i,j,k)-T(i-1,j,k))/dx)+v(i,j,k)*((T(i,j+1,k)-T(i,j,k))/dy)}};
end
end
end
% Graphical representation of the wave at different selected times
plot(x,u(:,1),'-',x,u(:,10),'-',x,u(:,50),'-',x,u(:,100),'-')
title('graphs')
xlabel('X')
ylabel('Y')
But I am getting this error
Subscript indices must either be real positive integers or logicals.
I am trying to implement this
with boundary conditions
Can someone please help me out!
Thanks

Answer 1

To be quite honest, it looks like you started with something that's way over your head, just typed everything down in one go without thinking much, and now you are surprised that it doesn't work...
In the future, please break down problems like these into waaaay smaller chunks that you can individually plot, check, test, etc. Better yet, try simpler problems first (wave equation, heat equation, ...), gradually working your way up to this.
I say this so harshly, because there were quite a number of fairly basic things wrong with your code:
you've used braces ({}) and brackets ([]) exactly as they are written in the equation. In MATLAB, braces are a constructor for a special container object called a cell array, and brackets are used to construct arrays and matrices. To group things like in the equation, you always have to use parentheses (()).
You had quite a number of parentheses wrong, which became apparent when I re-grouped and broke up those huge unintelligible lines into multiple lines that humans can actually read with understanding
you forgot to take the absolute values in the 3rd and 4th terms of u
you looped over k = 0:K and j = 1:m and then happily index everything with k and j-1. MATLAB is 1-based, meaning, the first element of anything is element 1, and indexing with 0 is an error
you've initialized 3 vectors u, v and T, but then index those in the loop as if they are 3D arrays
Now, I've managed to come up with the following code, which runs OK and at least more or less agrees with the equations shown. But I think it still doesn't make much sense because I get only zeros out (except for the initial values).
But, with this feedback, you should be able to correct any problems left.
Lmax = 1.0; % Maximum length
Wmax = 1.0; % Maximum wedth
Tmax = 2.; % Maximum time
% Parameters needed to solve the equation
K = 30; % Number of time steps
n = 3; % Number of space steps
m = 30; % Number of space steps
M = 2;
N = 1;
Pr = 1;
Re = 1;
Gr = 5;
maxn = 20; % The wave-front: intermediate point from which u=0
maxm = 20;
maxk = 20;
dt = Tmax/K;
dx = Lmax/n;
dy = Wmax/m;
%M = a*B1^2*l/(p*U)
b = 1/(1+M*dt);
c = dt/(1+M*dt);
d = dt/((1+M*dt)*dy);
%Gr = gB*(T-T1)*l/U^2;
% Initial value of the function u (amplitude of the wave)
u = zeros(n,m,K+1);
x = zeros(n,1);
for i = 1:n
if i < maxn
u(i,1)=1.;
else
u(i,1)=0.;
end
x(i) =(i-1)*dx;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
v = zeros(n,m,K+1);
y = zeros(m,1);
for j = 1:m
if j < maxm
v(1,j,1)=1.;
else
v(1,j,1)=0.;
end
y(j) =(j-1)*dy;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
T = zeros(n,m,K+1);
z = zeros(K,1);
for k = 1:K
if k < maxk
T(1,1,k)=1.;
else
T(1,1,k)=0.;
end
z(k) =(k-1)*dt;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Value at the boundary
%for k=0:K
%end
% Implementation of the explicit method
for k = 2:K % Time loop
for i = 2:n % Space loop
for j = 2:m-1
u(i,j,k+1) = b*u(i,j,k) + ...
c*Gr*T(i,j,k+1) + ...
d*(abs(u(i,j+1,k) - u(i,j ,k))/dy)^(N-1)*((u(i,j+1,k) - u(i,j ,k))/dy) - ...
d*(abs(u(i,j ,k) - u(i,j-1,k))/dy)^(N-1)*((u(i,j ,k) - u(i,j-1,k))/dy) - ...
d*(u(i,j,k)*((u(i,j ,k) - u(i-1,j,k))/dx) +...
v(i,j,k)*((u(i,j+1,k) - u(i ,j,k))/dy));
v(i,j,k+1) = dy*(u(i-1,j,k+1)-u(i,j,k+1))/dx + ...
v(i,j-1,k+1);
T(i,j,k+1) = T(i,j,k) + dt/(Pr*Re) * (...
(T(i,j+1,k) - 2*T(i,j,k) + T(i,j-1,k))/dy^2 - Pr*Re*(...
u(i,j,k)*((T(i,j,k) - T(i-1,j,k))/dx) + v(i,j,k)*((T(i,j+1,k) - T(i,j,k))/dy))...
);
end
end
end
% Graphical representation of the wave at different selected times
figure, hold on
plot(x, u(:, 1), '-',...
x, u(:, 10), '-',...
x, u(:, 50), '-',...
x, u(:,100), '-')
title('graphs')
xlabel('X')
ylabel('Y')