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) +...
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.
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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