Week5: added Error Plot

week5/ex1.py

@@ -42,32 +42,38 @@ def mk_phi_rk4(f):
 
     return phi_rk4
 
-
 # Multi Step
 #-----------
-def mk_phi_AB3(f, phi_short):
+def mk_phi_AB3(f, phi_short = None):
     steps = 3
     def phi_AB3(x, y, h):
-        if len(x) <= steps:
+        if len(x) < steps:
-            return phi_short(x, y, h)
+            if phi_short is None:
+                raise ValueError("This function needs at least {} steps, {} given.".format(steps, len(x)))
+            else:
+                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):
+def mk_phi_AB4(f, phi_short = None):
     steps = 4
     def phi_AB4(x, y, h):
-        if len(x) <= steps:
+        if len(x) < steps:
-            return phi_short(x, y, h)
+            if phi_short is None:
+                raise ValueError("This function needs at least {} steps, {} given.".format(steps, len(x)))
+            else:
+                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):
+def integrator(x, y_0, phi, y_i = None ):
     x = np.asarray(x)
     if isinstance(y_0, (list,np.ndarray)):
         y_0 = np.asarray(y_0)
@@ -80,8 +86,14 @@ def integrator(x, y_0, phi):
     y = np.zeros((N,M), dtype=np.float64)
     y[0] = y_0
 
+    j = 0
+    if y_i is not None:
+        for i, _ in enumerate(y_i):
+            y[i+1] = y_i[i]
+        j = i
+
     h = x[1]-x[0]
-    for i in range(1,N):
+    for i in range(j+1, N):
         y[i] = y[i-1] + h*phi(x[:i], y[:i], h)
 
     return y
@@ -104,7 +116,6 @@ def test_integrator_singlestep():
     exact_sol = ODEF_sol
 
     N = 2e1
-    M = 1
 
     # Domain on which to do the integration
     x = np.linspace(0, 1, N)
@@ -140,26 +151,35 @@ def test_integrator_multistep():
     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 = [ 
+    y_init = integrator(x[:4], y_0, mk_phi_rk4(test_func))
-            ["AB3", mk_phi_AB3],
+
-            ["AB4", mk_phi_AB4],
+    #Name, func, steps
-            ]
+    multi_schemes = [ 
+            ["AB3", mk_phi_AB3, 3],
+            ["AB4", mk_phi_AB4, 4],
+        ]
 
     # Show Plot
     from matplotlib import pyplot
     pyplot.subplots()
 
-    for name, func in schemes:
+    # Exact Solution
-        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")
+
+    # Plot the schemes
+    for name, func, steps in multi_schemes:
+        try:
+            pyplot.plot(x, integrator(x, y_0, func(test_func), y_init[:steps]), '--o', label=name)
+        except Exception as e:
+            print(e)
+            pass
+
     pyplot.xlabel("x")
     pyplot.ylabel("y")
 
@@ -167,7 +187,65 @@ def test_integrator_multistep():
 
     pyplot.show()
 
+def plot_integration_errors():
+    test_func = ODEF
+    exact_sol = ODEF_sol
+
+    # Domain on which to do the integration
+    x = np.arange(0, 2.001, step=0.02)
+
+    # Calculate Initial Values using RK4
+    y_0 = 0.5
+    y_init = integrator(x[:4], y_0, mk_phi_rk4(test_func))
+
+    single_schemes = [
+            ["Euler", mk_phi_euler],
+            ["Collatz", mk_phi_euler_mod],
+            ["Heun", mk_phi_heun],
+            ["RK4", mk_phi_rk4],
+
+        ]
+    multi_schemes = [ 
+            ["AB3", mk_phi_AB3, 3],
+            ["AB4", mk_phi_AB4, 4],
+        ]
+
+    # Show Plot
+    from matplotlib import pyplot
+    pyplot.subplots()
+
+    # Pre calculate the exact solution
+    exact = exact_sol(x)
+    exact = np.reshape(exact, (-1,1))
+
+    # Plot Single Step Schemes
+    for name, func in single_schemes:
+        try:
+            pyplot.plot(x, np.abs(exact - integrator(x, y_0, func(test_func))), '--o', label=name)
+        except Exception as e:
+            print(e)
+            pass
+
+    # Plot Multi Step Schemes
+    for name, func, steps in multi_schemes:
+        try:
+            pyplot.plot(x, np.abs(exact - integrator(x, y_0, func(test_func), y_init[:steps])), '--o', label=name)
+        except Exception as e:
+            print(e)
+            pass
+
+    pyplot.xlabel("x")
+    pyplot.ylabel("absolute error $|\\bar{y} - y|$")
+
+    pyplot.legend()
+
+    pyplot.show()
+
+
 if __name__ == "__main__":
     np.random.seed(0)
     #test_integrator_singlestep()
-    test_integrator_multistep()
+    #test_integrator_multistep()
+
+    plot_integration_errors()
+