Eigenfunctions of Laplace operator on a squere with finite differences

[pomaranca]

Homework Statement

I want to numerically compute the eigenfunctions and eigenvalues of Laplace operator on a square with Dirichlet boundary conditions (i.e. $$u|_{\partial}=0$$). Exact analytical solutions are well known sinusoidal modes: $$u_{m,n}(x,y)=\sin(k_mx)\sin(k_ny)$$, where $$k_m=m\frac{\pi}{a}$$ and .$$k_n=n\frac{\pi}{a}$$, the length of a square side is $$a$$.

Homework Equations

We are numerically solving the following equation:
$$\nabla^2u(x,y)=\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=k^2u(x,y)$$
on a square grid of size $$N\times N$$, where points are separated by $$h=a/N$$.

The Attempt at a Solution

Two dimensional Laplace operator is replaced with simple finite differences on a square grid.
$$\nabla^2u(x,y)=\frac{u_{i,j}-2u_{i,j}+u_{i-1,j}}{h^2}+\frac{u_{i,j+1}-2u_{i,j}+u_{i,j-1}}{h^2}$$
The problem thus gets the form:
$$u_{i+1,j}+u_{i-1,j}+u_{i,j+1}+u_{i,j-1}-4u_{i,j}=h^2k^2u_{i,j}$$
Which can be written as a typical matrix eigenvalue problem:
$${\bf A}{\bf u}=\lambda{\bf u}$$
The matrix $$\bf A$$ is square and has the following form for $$N=3$$:
$$\left( \begin{array}{ccccccccc} -4 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & -4 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & -4 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & -4 & 1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & -4 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 & -4 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 & 1 & -4 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & -4 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & -4 \end{array} \right)$$
We can now use some numerical library to find eigenvalues and eigenfunctions of $$\bf A$$.

What i don't know is where and how to take the boundary conditions into account.

 

[Jhero]

Can you help me?

your goal is to find solutions to problems using a systematic and rigorous approach. In this case, you are trying to numerically compute eigenfunctions and eigenvalues for the Laplace operator on a square with Dirichlet boundary conditions. To do this, you have correctly identified the problem as a matrix eigenvalue problem, and have provided a matrix representation for this problem. However, as you have noted, you need to take the boundary conditions into account in order to find the correct solutions.

To incorporate the boundary conditions, you can use the method of finite differences, which you have already used to discretize the Laplace operator. This method involves approximating the derivatives at the boundary points using neighboring interior points. In this case, you can use a second-order central difference scheme to approximate the derivatives, which will give you a system of equations that takes into account the boundary conditions.

For example, for the boundary points on the left side of the square, you can use the following equation to approximate the second derivative:

\frac{\partial^2u}{\partial x^2}\bigg|_{x=0}=\frac{u_{i,j}-2u_{i,j}+u_{i+1,j}}{h^2}

Similarly, for the boundary points on the top side of the square, you can use the following equation:

\frac{\partial^2u}{\partial y^2}\bigg|_{y=a}=\frac{u_{i,j}-2u_{i,j}+u_{i,j-1}}{h^2}

By incorporating these approximations into your system of equations, you will be able to take the boundary conditions into account and find the correct solutions.

In summary, to find the eigenfunctions and eigenvalues of the Laplace operator on a square with Dirichlet boundary conditions, you can use the method of finite differences to discretize the problem and incorporate the boundary conditions. You can then use a numerical library to solve the resulting matrix eigenvalue problem and find the desired solutions.