how to solve a 1d heat eqn with periodic BCs, u(0)=u(L) using implicit euler and finding the maximum error at Tend after comparing with the exact solution? obviously creating a finite difference matrix..i have done till finding the exact soln..

L = 1; %length of the rod
n = 2; 
nu = 1; % thermal diffusivity nu=1
Nx = 50; %length steps
Nt = 100; %time steps
Tend = 0.1;
mesh = linspace(0,L,Nx); %mesh size
dt = linspace(0,Tend,Nt); %time step interval
dx = mesh(2)-mesh(1); %length step interval

f = @(t,u) nu*(1/dx^2*A)*u;

lam = (n*pi/L)^2;
u(:,1) = sin(sqrt(lam)*mesh(1:Nx-1)); %initial condition

u_exact = exp^(-lam*t)*sin(sqrt(lam)*mesh(1:Nx-1));

[TJ]

I am not sure if I am right until now.. I have created the matrix A in a seperate function..

[TJ]

clear all;

L = 1; %length of the rod
n = 2; 
nu = 1; % thermal diffusivity nu=1
Nt = 50; %time steps
Nx = 600; %no of grid points
Tend = 0.1;
mesh = linspace(0,L,Nx); %mesh size
t = linspace(0,Tend,Nt); %time step interval
dt = t(2)-t(1);
dx = mesh(2)-mesh(1); %length step interval

f = @(t,u) nu*(1/dx^2*A)*u;

lam = (n*pi/L)^2;
u = zeros(length(mesh),Nx);
u(:,1) = sin(sqrt(lam)*mesh(1:end-1)); %initial condition
u_exact = exp(-lam*t)*sin(sqrt(lam)*mesh(1:end-1));

for k = 1:Nx-1
    %Implicit Euler
    t(k+1)=t(k)+dt;
    u(:,k+1) = u(:,k) + dt* f(t,u(:,k+1));
end
error1 = max(u_exact - u(:,end))/max(u_exact);
plot(Nx,error1);

[TJ]

    -2     1     0     0     1     0
     1    -2     1     0     0     0
     0     1    -2     1     0     0
     0     0     1    -2     1     0
     0     0     0     1    -2     1
     0     1     0     0     1    -2

this is how my A matrix is looking since u(0-1)=u(Nx-1)

[Walter Roberson]

Please be more specific about what your question is. What is it you are asking of us?

Answer 1

[Sebastian K]

For these types of computational problems, it is always a good idea to derive the mathematical equations first. This way you can split the problem into two parts:

1) Deriving the mathematical model.

2) Implementing your model in MATLAB.

I recommend that you first start with obtaining a clear form of the equations that you are trying to solve. It is important to understand the mathematics before you try to implement a code to solve a problem.

There are plenty of online course materials available regarding the solution of the 1-D heat equation using explicit and implicit methods. As an example consider the following slides:

Scientific Computing I by Michael Bader

Once you have a description of what equations you are trying to solve, people on this forum would be able to provide more useful recommendations.