Created
May 19, 2012 10:17
-
-
Save lobodin/2730359 to your computer and use it in GitHub Desktop.
Poisson equation solution using TDMA
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
| import math | |
| import matplotlib.pyplot as plt | |
| def tdma(a, b, c, d): | |
| ''' | |
| a[i] * x[i-1] + b[i] * x[i] + c[i] * x[i+1] = d[i] | |
| ''' | |
| length = len(a) | |
| for i in xrange(1, length): | |
| tmp = a[i] / b[i-1] | |
| b[i] = b[i] - tmp * c[i-1] | |
| d[i] = d[i] - tmp * d[i-1] | |
| x = a | |
| x[-1] = d[-1] / b[-1] | |
| for i in xrange(length - 2, -1, -1): | |
| x[i] = (d[i] - c[i] * x[i+1]) / b[i] | |
| return x | |
| def poisson_equation(h, x0, xn, a0, b0, c0, an, bn, cn, function): | |
| ''' | |
| Problem definition: | |
| Laplace(u) = f(x), x0 <= x <= xn | |
| a0 * du/dx + b0 * u(x) = c0, x = x0 | |
| an * du/dx + db * u(x) = cn, x = xn | |
| Linear equations system: | |
| (h * b0 - a0) * u0 + a0 * u1 = c0 * h; | |
| u[i-1] - 2u[i] + u[i+1] = h^2 * f[i], i = 1..n-1; | |
| -a0 * un + (an + h * bn) * u[n+1] = cn * h; | |
| ''' | |
| n = int((xn - x0) / h) | |
| a, b, c, d = [], [], [], [] | |
| for i in xrange(n + 1): | |
| a.append(1) | |
| b.append(-2) | |
| c.append(1) | |
| d.append(h * h * function(x0 + i * h)) | |
| a[n] = -an | |
| b[0] = h * b0 - a0 | |
| b[n] = an + h * bn | |
| c[0] = a0 | |
| d[0] = c0 * h | |
| d[n] = cn * h | |
| u = tdma(a, b, c, d) | |
| return u | |
| def test(): | |
| def example_function(x): | |
| return math.sin(x) | |
| def exact_solution(i, h, x0): | |
| x = x0 + i * h | |
| return -math.sin(x) - 0.0116 * x + 1.7082 | |
| def plot(u, exact_u): | |
| plt.plot(range(len(u)), u) | |
| plt.plot(range(len(exact_u)), exact_u) | |
| plt.show() | |
| h = 0.1 # step | |
| x0 = 0.0 # left border | |
| xn = 20.0 # right border | |
| a0, b0, c0, an, db, cn = 0.7, 1.0, 1.0, 0.3, 2.0, 1.0 # coefficients of boundary values | |
| u = poisson_equation(h, x0, xn, a0, b0, c0, an, db, cn, example_function) | |
| exact_u = map(lambda i: exact_solution(i, h, x0), range(len(u))) | |
| plot(u, exact_u) | |
| test() |
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment