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 2016-11-19

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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 2016-11-20

编辑：Walter Roberson 2016-11-20

在 MATLAB Online 中打开

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

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

Walter Roberson 2016-11-20

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

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Answer 1

Sebastian K 2016-11-23

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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.