Week5: Wrap up: plot accuracy/stepsize (needs fix)

week5/ex1.py

@@ -75,6 +75,7 @@ def mk_phi_AB4(f, phi_short = None):
 ##############
 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)
     else:
@@ -236,16 +237,85 @@ def plot_integration_errors():
 
     pyplot.xlabel("x")
     pyplot.ylabel("absolute error $|\\bar{y} - y|$")
+    pyplot.title("absolute error between an approach and the exact solution")
 
     pyplot.legend()
 
     pyplot.show()
 
 
+def accuracy_per_stepsize( debug=True ):
+    stepsizes = np.asarray([1e-1, 1e-2, 1e-3, 1e-5])
+
+    # define the problem to be solved
+    test_func = lambda x,y: x
+    exact_sol = lambda x: np.exp(x)
+
+    # set domain and intial value
+    x_min = 0
+    x_max = 2
+    y_0 = 1
+
+    exact_end = exact_sol(x_max)
+
+    # define the schemes to use
+    schemes = [
+            ["Euler", mk_phi_euler, 1],
+            ["Collatz", mk_phi_euler_mod, 1],
+            ["Heun", mk_phi_heun, 1],
+            ["RK4", mk_phi_rk4, 1],
+            ["AB3", mk_phi_AB3, 3],
+            ["AB4", mk_phi_AB4, 4],
+        ]
+
+    # get max steps needed for the multistep schemes
+    max_steps = 0
+    for _, _, steps in schemes:
+        if steps > max_steps:
+            max_steps = steps
+
+    # pre calculate for multistep integrations
+    y_init = np.zeros((len(stepsizes),1))# Note the dimensionality of y_0
+    max_steps += 1
+    for i, h in enumerate(stepsizes):
+        x = x_min + np.linspace(0, h * max_steps, max_steps, True)
+        # Note the dimensionality of y_0
+        # TODO: fix up this ugliness
+        z = integrator(x[:4], y_0, mk_phi_rk4(test_func))
+        y_init[i] = z[1,:]
+
+    # plot schemes
+    from matplotlib import pyplot
+    pyplot.subplots()
+    for name, scheme, steps in schemes:
+        diffs = np.zeros(len(stepsizes))
+        if debug:
+            print("Calculating {}".format(name))
+
+        for i, h in enumerate(stepsizes):
+            N = int(abs(x_max-x_min)/h)
+            x = np.linspace(x_min, x_max, N, True)
+            y = integrator(x, y_0, scheme(test_func), y_init[:steps])
+            diffs[i] = np.abs(exact_end - [-1])
+        
+        pyplot.plot(stepsizes, diffs, '--o', label=name)
+
+    pyplot.xlabel('Stepsize $h$')
+    pyplot.ylabel('Absolute Error')
+    pyplot.title('Absolute error at the end of integration')
+
+    pyplot.legend()
+
+    pyplot.show()
+
+
+def pendulum_problem():
+    pass
+
 if __name__ == "__main__":
     np.random.seed(0)
-    #test_integrator_singlestep()
+    test_integrator_singlestep()
     #test_integrator_multistep()
 
-    plot_integration_errors()
-
+    #plot_integration_errors()
+    #accuracy_per_stepsize()