How Can I Optimize My 2D Cylindrical Heat Equation Program?

[range.rover]

Hello friends, I am new for numerical methods and programming. i have been trying to devolop a program in 2D poisson heat equation in cylinder (r,angle) by finite difference method
∂²u/∂r² + 1/r * ∂u/∂r + 1/r² * ∂²u/∂θ² = Q(u,θ)discritized equation :-

u_i+1,j − 2u_ij + u_i−1,j/(∆r)² + 1/r_i * u_i+1,j − u_i−1,j/2∆r +1/r²_i * u_i,j+1−2u_ij +u_i,j−1/(∆θ)² = Q(u,θ)

i have been trying to develop this program, can anybody help me in making it to run faster for n = 400;

clear all
close all
clc
tic
n = 5; % mesh size
s = n-2;
r = linspace(1,5,n)-1;

angle =((1:1:n)-1)/(n)*2*pi;

m = (s*n); %number of nodesdr = r(3)-r(2); % distance between radius

dangle = angle(3)- angle(2); % change of angle between radius

Q = 20*dr^2; % heat sourse

% right hand side Matrix

rhs = zeros(m+1,1);
rhs( : ) = Q;
bou = ones(m+1,1);

% boundary values
bou( : ) = 0;
tic
for i = n-1:n-2:m+1
rhs(i,1) = rhs(i,1) + bou(i);
end
toc% evaluating the co effitiants of tridiagonal sparse matrix

tic
for i = 2:n-1
a(i) = 2*(1 + ((dr^2)/(r(i)*dangle)^2));
b(i) = 1+(1/(2*r(i)));
c(i) = 1-(1/(2*r(i)));
d(i) = dr^2/(r(i)^2*dangle^2);
end
toc
% building up the matrix for A.x = B

f = c(2);
I = speye(n);
a = (a(2:end))';
b = [0;b(2:n-2)'];
p = [c(3:end)';0];
% d = (d(2:end))';
T = spdiags([p -1*a b],[-1 0 1],n-2,n-2);
oops = full(T);
FirstM = kron(I,T);
% z = full(FirstM);
d = (d(2:end));
e = repmat(d,1,n-1);
tic
for i = 1: (m-s)
FirstM(i,(i+s)) = e(i);

FirstM((i+s),i) = e(i);
end
toc
tic
for i = 1: s
FirstM((m-s+i),i) = e(i);
FirstM(i,(m-s+i)) = e(i);
end
toc
% z = full(FirstM);

FirstM(1,m+1) = 0;
FirstM(m+1,1) = 0;
tic
for i = 1:n-2:m+1
FirstM(i,m+1) = f;
FirstM(m+1,i) = 1;
end
toc
FirstM(m+1,m+1) = -4;
z = full(FirstM);

% solving A*x = B for solution
Tsol = abs(mldivide(FirstM,rhs));
toc
T1soln = Tsol(1:m,1);
Tsoln = reshape(Tsol,n-1,n-1);
% Tsoln = Tsoln';% Tsoln = reshape(T1soln,n,s);
% angle = angle(1,2:n);
% r = r(1,2:n);
[x,y] = meshgrid(r,angle);
[xs,ys] = pol2cart(x,y);
surf(xs,ys,Tsoln,'EdgeColor','none').

 

[gtoguy97]

I can suggest some methods to help you make the program run faster. Firstly, you should try to use vectorized operations instead of using for loops whenever possible. Secondly, you can try to optimize your code by avoiding unnecessary computations or data accesses. Finally, you can try to use parallelization techniques such as OpenMP or MPI to make your program run faster.