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..

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TJ
TJ 2016-11-19
回答： Sebastian K 2016-11-23
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));
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TJ
TJ 2016-11-20
编辑：Walter Roberson 2016-11-20
-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)

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Sebastian K
Sebastian K 2016-11-23
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.

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