What is the second-order approximation error of the Heun method, which averages the slope at the beginning and the end of the interval?
deconstructing the Heun's method:
\begin{equation} \underbrace{y(x + h)}_{\text{output}} = \underbrace{y(x)}_{\text{input}} + \frac {h}{2} \{ \underbrace{f(x,y(x))}_{\text{start slope}} + \underbrace{f[x + h, y(x) + hf(x,y(x))]}_{\text{end slope}} \} \end{equation}expanding the $\text{end slope}$:
plugging it back into the first equation:
This equation matches the polynomial expansion of the solution of the differential equation upto second order; this method will give an answer that is correct to the second order.
For a simple harmonic oscillator $\ddot y + y = 0$, with initial conditions $y(0) = 1, \dot y(0) = 0$, find $y(t)$ from $t = 0$ to $100 \pi $. Use an Euler method and a fixed-step fourth-order Runge-Kutta method. For each method check how the average local error, and the error in the final value and slope, depend on the step size.
Euler's method at time $t$ and stepsize $h$:
\begin{equation}
y(t+h) = y(t) + h\dot y(t) \\[3pt]
\dot y(t+h) = \dot y(t) + h\ddot y(t)
\end{equation}
Since $\ddot y = - y$ for harmonic oscillator,
import numpy as np
import matplotlib.pyplot as plt
import math
plt.rcParams['figure.figsize'] = 14, 6
def plot_y(f, t_steps, y_0, dy_0):
y_plot = f(t_steps, y_0, dy_0)[0]
plt.plot(t_steps, y_plot)
plt.xlabel("t")
plt.ylabel("y")
plt.show()
def plot_dy(f, t_steps, y_0, dy_0):
dy_plot = f(t_steps, y_0, dy_0)[1]
plt.plot(t_steps, dy_plot)
plt.xlabel("t")
plt.ylabel("dy")
plt.show()
t_start = 0
t_end = 100*math.pi
h = 0.01
t_steps = np.arange(t_start, t_end, h)
print(len(t_steps))
y_0 = 1
dy_0 = 0
def euler_method(t_steps, y_0, dy_0):
y_t = y_0
dy_t = dy_0
y = []
dy = []
for t_now in t_steps:
y.append(y_t)
dy.append(dy_t)
tmp = dy_t
dy_t = dy_t - h*y_t
y_t = y_t + h*tmp
return np.array(y), np.array(dy)
plot_y(euler_method, t_steps, y_0, dy_0)
plot_dy(euler_method, t_steps, y_0, dy_0)
31416
def rk_step(h, (y, dy)):
k1 = [0,0]
k2 = [0,0]
k3 = [0,0]
k4 = [0,0]
k1[0] = h * dy # position change @ y
k1[1] = -h * y # slope change @ y
k2[0] = h * (dy + k1[1]/2) # position change @ y + k1/2
k2[1] = -h * (y + k1[0]/2) # slope change @ y + k1/2
k3[0] = h * (dy + k2[1]/2) # position change @ y + k2/2
k3[1] = -h * (y + k2[0]/2) # slope change @ y + k2/2
k4[0] = h * (dy + k3[1]) # position change @ y + k3
k4[1] = -h * (y + k3[0]) # slope change # y + k3
y_new = y + (k1[0] + 2*k2[0] + 2*k3[0] + k4[0]) / 6
dy_new = dy + (k1[1] + 2*k2[1] + 2*k3[1] + k4[1]) / 6
return y_new, dy_new
def runge_kutta(t_steps, y_0, dy_0):
y_t =1
dy_t = 0
y = []
dy = []
step = t_steps[1]- t_steps[0]
for t_now in t_steps:
y.append(y_t)
dy.append(dy_t)
y_t, dy_t = rk_step(step, (y_t, dy_t))
return np.array(y), np.array(dy)
plot_y(runge_kutta, t_steps, y_0, dy_0)
plot_dy(runge_kutta, t_steps, y_0, dy_0)
Assuming $ y = cos(t) $ from $y(0) = 1$ and $\dot y(0) = 0$:
def actual(t_steps, y_0, dy_0):
y = []
dy = []
for t_now in np.arange(t_start, t_end, h):
y.append(math.cos(t_now))
dy.append(math.sin(t_now))
return np.array(y), np.array(dy)
plot_y(actual, t_steps, y_0, dy_0)
plot_dy(actual, t_steps, y_0, dy_0)
Error Estimate goes here.
7.3
def runge_kutta_adaptive(h_0, t_end, y_0, dy_0, threshold):
y_t =1
dy_t = 0
h_t = h_0
t_now = 0
y = []
dy = []
while t_now < t_end:
y.append(y_t)
dy.append(dy_t)
y_t1, dy_t1 = rk_step(h_t, (y_t, dy_t))
y_t2, dy_t2 = rk_step(h_t/2, rk_step(h_t/2, (y_t, dy_t)))
err_est = abs(y_t - y_t2)
print(err_est)
print("t:" + str(t_now))
if err_est > threshold:
h_t *= .5
elif err_est <= threshold:
h_t *= 2
t_now += h_t
y_t, dy_t = rk_step(h_t, (y_t, dy_t))
return np.array(y), np.array(dy)
def plot_y(f, h_0, t_end, y_0, dy_0, threshold):
y_plot = f(h_0, t_end, y_0, dy_0, threshold)[0]
plt.plot(y_plot)
plt.xlabel("t")
plt.ylabel("y")
plt.show()
def plot_dy(f, h_0, t_end, y_0, dy_0, threshold):
dy_plot = f(h_0, t_end, y_0, dy_0, threshold)[1]
plt.plot(y_plot)
plt.xlabel("t")
plt.ylabel("dy")
plt.show()
threshold = 0.01
h_0 = 0.1
t_end = 100*math.pi
plot_y(runge_kutta_adaptive, h_0, t_end, y_0, dy_0, threshold)
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