Week5: upto c)

week5/ex1.py

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+#!/usr/bin/env python3
+
+"""Ordinary Differential Equations"""
+
+import numpy as np
+
+
+def mk_phi_euler(f):
+    """ Return a Phi computed for the Euler Method. """
+    def phi_euler(x, y, h):
+        return f(x[-1], y[-1])
+
+    return phi_euler
+
+
+def mk_phi_euler_mod(f):
+    """ Return a Phi computed for the Modified Euler (Collatz) Method. """
+    def phi_euler_mod(x, y, h):
+        return f(x[-1] + 0.5*h, y[-1] + 0.5*h*f(x[-1], y[-1]))
+
+    return phi_euler_mod
+
+
+def mk_phi_heun(f):
+    """ Return a Phi computed for the Heun Method. """
+    def phi_heun(x, y, h):
+        return (f(x[-1], y[-1]) + f(x[-1] + h, y[-1] + h * f(x[-1], y[-1])))/2
+    
+    return phi_heun
+
+def mk_phi_rk4(f):
+    """ Return a Phi computed for the 4th order Runge-Kutta Method. """
+    def phi_rk4(x, y, h):
+        k1 = f(x[-1], y[-1])
+        k2 = f(x[-1] + 0.5*h, y[-1] + 0.5*k1)
+        k3 = f(x[-1] + 0.5*h, y[-1] + 0.5*h*k2)
+        k4 = f(x[-1] + h, y[-1] + h*k3)
+
+        return (k1 + 2*k2 + 2*k3 + k4)/6
+
+    return phi_rk4
+
+def integrator(x, y_0, phi):
+    x = np.asarray(x)
+    if isinstance(y_0, (list,np.ndarray)):
+        y_0 = np.asarray(y_0)
+    else:
+        y_0 = np.array([ y_0 ])
+
+    N = len(x)
+    M = len(y_0)
+
+    y = np.zeros((N,M), dtype=np.float64)
+    y[0] = y_0
+
+    h = x[1]-x[0]
+    for i, x_i in enumerate(x):
+        # Skip the first iteration
+        if not i:
+            continue
+
+        y[i] = y[i-1] + h*phi(x[:i], y[:i], h)
+
+    return y
+
+
+
+def test_integrator():
+    np.random.seed(0)
+
+    def ODEF(x, y):
+        return y - x**2 + 1
+    
+    def ODEF_sol(x):
+        return (x+1)**2 - 0.5*np.exp(x)
+
+
+    test_func = ODEF
+    exact_sol = ODEF_sol
+
+    N = 2e1
+    M = 1
+
+    # Domain on which to do the integration
+    x = np.linspace(0, 1, N)
+
+    # Generate Initial Value
+    y_0 = 0.5
+
+    schemes = [ 
+            ["Euler", mk_phi_euler],
+            ["Collatz", mk_phi_euler_mod],
+            ["Heun", mk_phi_heun],
+            ["RK4", mk_phi_rk4],
+            ]
+
+    # Show Plot
+    from matplotlib import pyplot
+    pyplot.subplots()
+
+
+    for name, func in schemes:
+        pyplot.plot(x, integrator(x, y_0, func(test_func)), '--o', label=name)
+
+    pyplot.plot(x, exact_sol(x), '-', label="Exact Solution")
+    pyplot.xlabel("x")
+    pyplot.ylabel("y")
+
+    pyplot.legend()
+
+    pyplot.show()
+
+if __name__ == "__main__":
+    test_integrator()