Week5: Wrap up: MultiStep

week5/ex1.py

@@ -4,7 +4,11 @@
 
 import numpy as np
 
+# Integrations Schemes #
+########################
 
+# Single Step
+#------------
 def mk_phi_euler(f):
     """ Return a Phi computed for the Euler Method. """
     def phi_euler(x, y, h):
@@ -12,7 +16,6 @@ def mk_phi_euler(f):
 
     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):
@@ -20,7 +23,6 @@ def mk_phi_euler_mod(f):
 
     return phi_euler_mod
 
-
 def mk_phi_heun(f):
     """ Return a Phi computed for the Heun Method. """
     def phi_heun(x, y, h):
@@ -40,6 +42,31 @@ def mk_phi_rk4(f):
 
     return phi_rk4
 
+
+# Multi Step
+#-----------
+def mk_phi_AB3(f, phi_short):
+    steps = 3
+    def phi_AB3(x, y, h):
+        if len(x) <= steps:
+            return phi_short(x, y, h)
+        else:
+            return ( 23*f(x[-1], y[-1]) - 16*f(x[-2], y[-2]) + 5*f(x[-3], y[-3]) )/12
+
+    return phi_AB3
+
+def mk_phi_AB4(f, phi_short):
+    steps = 4
+    def phi_AB4(x, y, h):
+        if len(x) <= steps:
+            return phi_short(x, y, h)
+        else:
+            return ( 55*f(x[-1], y[-1]) - 59*f(x[-2], y[-2]) + 37*f(x[-3], y[-3]) -9*f(x[-4],y[-4]))/24
+
+    return phi_AB4
+
+# Integrator #
+##############
 def integrator(x, y_0, phi):
     x = np.asarray(x)
     if isinstance(y_0, (list,np.ndarray)):
@@ -54,26 +81,24 @@ def integrator(x, y_0, phi):
     y[0] = y_0
 
     h = x[1]-x[0]
-    for i, x_i in enumerate(x):
+    for i in range(1,N):
-        # Skip the first iteration
-        if not i:
-            continue
-
         y[i] = y[i-1] + h*phi(x[:i], y[:i], h)
 
     return y
 
 
+# Math Test Functions #
+#######################
+def ODEF(x, y):
+    return y - x**2 + 1
 
-def test_integrator():
+def ODEF_sol(x):
-    np.random.seed(0)
+    return (x+1)**2 - 0.5*np.exp(x)
 
-    def ODEF(x, y):
-        return y - x**2 + 1
-    
-    def ODEF_sol(x):
-        return (x+1)**2 - 0.5*np.exp(x)
 
+# Testing #
+###########
+def test_integrator_singlestep():
 
     test_func = ODEF
     exact_sol = ODEF_sol
@@ -110,5 +135,39 @@ def test_integrator():
 
     pyplot.show()
 
+def test_integrator_multistep():
+    test_func = ODEF
+    exact_sol = ODEF_sol
+
+    N = 2e1
+    M = 1
+
+    # Domain on which to do the integration
+    x = np.linspace(0, 1, N)
+
+    # Calculate Initial Values using RK4
+    y_0 = 0.5
+    schemes = [ 
+            ["AB3", mk_phi_AB3],
+            ["AB4", mk_phi_AB4],
+            ]
+
+    # Show Plot
+    from matplotlib import pyplot
+    pyplot.subplots()
+
+    for name, func in schemes:
+        pyplot.plot(x, integrator(x, y_0, func(test_func, mk_phi_rk4(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()
+    np.random.seed(0)
+    #test_integrator_singlestep()
+    test_integrator_multistep()