ans =
Numerical solution for the heat equation does not match the exact solution
显示 更早的评论
I write a matlab code for the problem, and i want to compare the numerucal and the exact solution at t=0.5 but the graph of the soultions are very different.
The problem is in the following:
% Parameters
L = 5; % Length of the rod
T = 1; % Total time
Nx = 100; % Number of spatial points
Nt = 500; % Number of time steps
alpha = 0.01; % Thermal diffusivity
% Discretization
dx = 0.1; % Spatial step size
dt = 0.04; % Time step size
r = alpha * dt / dx^2; % Stability parameter for the scheme
% Initial condition
x = linspace(0, L, Nx);% generates Nx points. The spacing between the points is (L-0)/(Nx-1).
u = cos(x-0.5); % initial condition
% Preallocate solution matrix
U = zeros(Nx, Nt+1); % returns an Nx-by-Nt matrix of zeros
U(:,1) = u; %extracts all the elements from the first column of matrix
% Time-stepping loop
for n = 1:Nt
% Update the solution using explicit scheme for the Heat Equation
for i = 2:Nx-1
U(i,n+1) = U(i,n) + r * (U(i+1,n) - 2*U(i,n) + U(i-1,n));
end
end
Numerical_sol = U
% Plot results
t = linspace(0, T, Nt+1);
[X, T] = meshgrid(x, t);
% Exact solution
Exact= @(x, t) exp(-t) .* cos(pi * x - 0.5 * pi);
figure;
mesh(X, T, U');
xlabel('x');
ylabel('t');
zlabel('u(x,t)');
title('Heat Equation Solution');
figure
plot(x, U(:,1), 'o');
2 个评论
John D'Errico
2024-9-5
You accepted the answer to this question. Then you asked it just again, 42 minutes ago. I closed the duplicate question, since it asks nothing you should not have learned here.
Torsten
2024-9-5
I didn't know that you are the Website Master taking the right to delete questions and answers that are actually in flow.
采纳的回答
I cannot believe that the value for alpha has no influence on the solution.
So I guess that the exact solution maybe holds for alpha = 1.
10 个评论
Erm
2024-9-4
Why don't you use the values for L, dx and dt from your assignment ?
% Parameters
L = 1; % Length of the rod
T = 1; % Total time
% Discretization
dx = 0.1; % Spatial step size
dt = 0.04; % Time step size
Nx = L/dx + 1; % Number of spatial points
Nt = T/dt; % Number of time steps
alpha = 1; % Thermal diffusivity
r = alpha * dt / dx^2; % Stability parameter for the scheme
% Initial condition
x = linspace(0, L, Nx);% generates Nx points. The spacing between the points is (L-0)/(Nx-1).
u = cos(pi*(x-0.5)); % initial condition
% Preallocate solution matrix
U = zeros(Nx, Nt+1); % returns an Nx-by-Nt matrix of zeros
U(:,1) = u; %extracts all the elements from the first column of matrix
% Time-stepping loop
for n = 1:Nt
% Update the solution using explicit scheme for the Heat Equation
for i = 2:Nx-1
U(i,n+1) = U(i,n) + r * (U(i+1,n) - 2*U(i,n) + U(i-1,n));
end
end
plot(x,[U(:,1),U(:,2),U(:,3),U(:,4),U(:,5)])
Erm
2024-9-4
Torsten
2024-9-4
The solution you gave is not the exact solution.
Just differentiate exp(-t)*cos(pi*(x-0.5)) with respect to t and two times with respect to x and see that the results are different.
Erm
2024-9-4
Erm
2024-9-4
Erm
2024-9-4
Torsten
2024-9-4
The blue dotted curve cannot be correct because u(0,t) = u(1,t) = 0 must hold for all times t.
And as said:
The solution you call "exact" is not a solution of the differential equation. Thus comparing with the numerical solution makes no sense.
The exact solution for the differential equation
du/dt = alpha * d^2u/dx^2
with initial condition
u(0,x) = cos(pi*(x-0.5))
and boundary conditions
u(t,0) = u(t,1) = 0
is given by
uexact(t,x) = exp(-alpha*pi^2*t) * cos(pi*(x-0.5))
That's the function you have to compare the numerical solution with.
syms x t alpha pi
u = exp(-alpha*pi^2*t) * cos(pi*(x-0.5));
diff(u,t)
alpha*diff(u,x,2)
Erm
2024-9-4
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