ex4

ex4/code/task_c.py

@@ -0,0 +1,61 @@
+import numpy as np
+import matplotlib.pyplot as plt
+
+# Peclet problem:
+#  -u''(x) + pu'(x) = 0    x in (0,1)
+#              u(0) = 0
+#              u(1) = 1
+
+# parameters
+p = 70
+n_values = [10,20,30,40,70]       # elements
+
+# exact solution
+x_exact = np.linspace(0,1,1000)
+u_exact = (np.exp(p*x_exact)-1)/(np.exp(p)-1)
+
+for n in n_values:
+    # mesh
+    m = n+1      # nodes
+    h = 1.0/n
+    x = np.linspace(0,1,m)
+
+    # local stiffness matrix
+    K_loc = np.zeros((2,2))
+    A = (1.0/h) * np.array([[ 1,-1], [-1, 1]])
+    B = (p/2) * np.array([[ -1, 1], [ -1, 1]])
+    K_loc = A+B
+
+    # Assembling
+    K = np.zeros((m,m))
+    F = np.zeros(m)
+    for i in range(n):
+        K[i:i+2,i:i+2] += K_loc
+
+    # Boundary conditions
+    # Dirichlet: u(0) = 0
+    K[0,:] = 0
+    K[0,0] = 1
+    F[0]   = 0   
+    # Dirichlet: u(1) = 1
+    K[-1,:] = 0
+    K[-1,-1] = 1
+    F[-1]   = 1
+
+    u = np.linalg.solve(K, F)
+    plt.plot(x, u, "-o", markersize=2, label=f"n = {n}")
+
+plt.plot(x_exact, u_exact, "black", label="exact")
+plt.title(f"p = {p}")
+plt.xlabel("x")
+plt.ylabel("u(x)")
+plt.legend()
+plt.grid(True)
+
+print("K = ", K)
+print("f = ", F)
+print("u = ", u)
+
+plt.tight_layout()
+plt.savefig("../task_c.png", dpi=300)
+plt.show()