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Finite Difference Matrix Help

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Justin Yeung
Justin Yeung on 16 Feb 2020
Edited: Fabio Freschi on 21 Feb 2020
So I have a finite difference problem with beam bending. I am trying define a matrix that follows the 4th order ODE for a Central Difference formula.
n= 10 %number of nodes in the beam
%% Step 2: Define the A Matrix
A = (((2*eye(N) +...
diag(ones(N-1,1),1)))+...
diag(ones(N-1,1),-1));
Now this generates a matrix of size N with '2' along the main diagonal. However, a 4th order ODE is different. So I guess my question is, is there a way to add in values '2' off the main diagonal? I've been trying to mess around with the code above, but it keeps saying the "matrix size dimensions must agree.

  2 Comments

Sindar
Sindar on 16 Feb 2020
check out spdiags. The first example creates the second order second derivative
Srivardhan Gadila
Srivardhan Gadila on 20 Feb 2020
@Justin Yeung are you looking for a matrix which has zeros as diagonal elements, 2's as non-diagonal elements and add this matrix to some other matrix? If not, can you please be more specific or can you give an example?

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Answers (1)

Fabio Freschi
Fabio Freschi on 20 Feb 2020
Edited: Fabio Freschi on 21 Feb 2020
Edit: I changed my answer including a reference and the second order derivative
The coefficients for central differences of different order of accuracy with uniform spacing can be found on wikipedia here.
I assume you need a second order derivative. If it is the case, you can build the matrix using spdiags:
N = 10;
% coefficients (Derivative 2, Accuracy 4 of the wikipedia table)
C = [ones(N,1)/12 4*ones(N,1)/3 -5*ones(N,1)/2 4*ones(N,1)/3 ones(N,1)/12];
% positions along the diagonal
idiag = -2:2;
% matrix
A = spdiags(C,idiag,N,N);
Remember to divide the matrix by the step size dx^2.
The matrix created in this way is sparse (as it is usually done with these problems).

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