Is it possible to use finite difference methods to solve a 2-dimensional wave equation (PDE) with boundary conditions? I know finite difference methods can be used to solve a 1-d wave equation, but how would this be changed for 2-dimensions (or is it even possible)? (I want to find the frequencies of vibration of a drum)
2026-04-01 14:27:56.1775053676
Finite difference method for a PDE in 2-dimensions?
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The standard finite difference scheme for 2D wave equation is $$\frac{U(i,j,k+1) - 2U(i,j,k)+U(i,j,k-1)}{(\Delta t)^2} = \frac{U(i+1,j,k) - 2U(i,j,k)+U(i-1,j,k)}{(\Delta x)^2} + \frac{U(i,j+1,k) - 2U(i,j,k)+U(i,j-1,k)}{(\Delta x)^2}$$ where the variables $(x_1,x_2,t)$ correspond to the indices $i,j,k$. (This is copied from a Computational Science post by Paul).
However, if you are looking for eigenvalues of Laplacian, that is not the same as solving wave equation numerically. This problem is more painful with finite differences, especially when the domain is not rectangular. You have to work quite a bit to set up a large (though sparse) matrix of the discrete Laplacian and then hope your linear solver can handle the eigenvalue problem.
Generally, one would use FEM for this. See section 1.6 of Lecture Notes on Solving Large Scale Eigenvalue Problems by Peter Arbenz.