2d wave equation with finite differences blowing up
I am (naively) trying to solve the 2d wave equation with finite differences. But the system blows up instantly.
For simplicity I set the constant $c=1$, then I am left with $$\Delta u =u_{tt}.$$
I set up the FDM with
N = 9
u = np.zeros((N**2,1))
u.reshape(N,N)[N//2,N//2] = 1
Now, at the time t=0 the function looks like this
u_0 =
[[0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0]
[0 0 0 0 1 0 0 0 0]
[0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0]]
To which I apply the five point diffence stencil Lh
import scipy.sparse as sp
import numpy as np
h = 1/(N+1)
T = np.diag(N*[-2]) + np.diag((N-1)*[1],k=-1) +np.diag((N-1)*[1],k=1)
spt = sp.coo_matrix(T)
spi = sp.eye(N)
Lh = (sp.kron(spi,spt) + sp.kron(spt,spi) ) / h**2
Since Lh is my discretized Laplacian, for every timestep I add the acceleration that it gives me to some book-keeping vector v to change the velocity accordingly:
v = np.zeros_like(u)
print(u.reshape(N,N).astype(int))
for i in range(3):
v += Lh.dot(u)
u += v
print(u.reshape(N,N).astype(int))
Here is what happens:
[[ 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 99 0 0 0 0]
[ 0 0 0 99 -398 99 0 0 0]
[ 0 0 0 0 99 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0]]
In the next timestep:
[[ 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 9999 0 0 0 0]
[ 0 0 0 19999 -<PHONE_NUMBER> 0 0 0]
[ 0 0 9999 -79699 198800 -79699 9999 0 0]
[ 0 0 0 19999 -<PHONE_NUMBER> 0 0 0]
[ 0 0 0 0 9999 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0]]
What is the problem here? I have not set up a boundary condition to my solution. Maybe I should?
greetings 935