Finite Differences for Boundary Value Problems

In [6]:
import numpy as np
import matplotlib.pyplot as pt

import scipy.sparse as sps

We'll solve

$u''+1000(1+x^2)u=0$ on $(-1,1)$

with $u(-1)=3$ and $u(1)=-3$.

In [20]:
#n = 9
n = 200

mesh = np.linspace(-1, 1, n)
h = mesh[1] - mesh[0]

Use sps.diags(values, offsets=..., shape=(n, n)) to make a centered difference matrix.

In [30]:
A = sps.diags(
    [1,-2,1],
    offsets=[-1,0,1], 
    shape=(n, n))

if n < 10:
    print(A.todense())

Create second_deriv as a matrix to apply the second derivative. Can only do that for the interior points!

  • change shape and offsets
  • Take h into account
In [31]:
second_deriv = sps.diags(
    [1,-2,1],
    offsets=np.array([-1,0,1])+1,
    shape=(n-2, n))/h**2


if n < 10:
    print(second_deriv.todense())

Make a matrix for the lower-order term.

In [32]:
factor = sps.diags(
    [1000*(1 + mesh[1:]**2)],
    offsets=[1],
    shape=(n-2, n))

if n < 10:
    print(mesh[1:-1])
    print()
    print(factor.todense())

Build the matrix for the interior:

In [24]:
A_int = second_deriv+factor

if n < 10:
    print(A_int.todense())

Glue on the rows for the boundary conditions:

In [25]:
A = sps.vstack([
    sps.coo_matrix(([1], ([0],[0])), shape=(1, n)),
    A_int,
    sps.coo_matrix(([1], ([0],[n-1])), shape=(1, n)),
    ])
A = sps.csr_matrix(A)

if n < 10:
    print(A.todense())

Next, assemble the right-hand side as rhs:

Pay special attention to the boundary conditions. What entries of rhs do they correspond to?

In [26]:
rhs = np.zeros(n)
rhs[0] = 3
rhs[-1] = -3

To wrap up, solve and plot:

In [27]:
import scipy.sparse.linalg as sla

sol = sla.spsolve(A, rhs)
In [28]:
pt.plot(mesh, sol)
Out[28]:
[<matplotlib.lines.Line2D at 0x7fee957500f0>]
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