3 Finite Differences – ODE


7.1


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 $\text{end slope}$:

\begin{equation} \begin{aligned} & f[x + h, y(x) + hf(x,y(x))] \\[3pt] = \; & f(x,y(x)) + h \frac{d}{dh} f[x+h, y(x) + hf(x,y(x))]_{h\,=\,0} + \mathcal{O}(h^2) \\[3pt] = \; & f(x,y(x)) + h \left[ \frac{\delta f}{\delta x} + f(x,y(x)) \frac{\delta f}{\delta y} \right] + \mathcal{O}(h^2) \end{aligned} \end{equation}


plugging it back into the first equation:

\begin{equation} \begin{aligned} y(x+h) &= y(x) + \frac{h}{2} \left\{ 2f(x,y(x)) + h\left[ \frac{\delta f}{\delta x} + f(x,y(x)) \frac{\delta f}{\delta y}\right] + \mathcal{O}(h^2)\right\} \\[3pt] &= y(x) + hf(x,y(x)) + \frac{h^2}{2}\left[ \frac{\delta f}{\delta x} + f(x,y(x)) \frac{\delta f}{\delta y}\right] + \mathcal{O}(h^3) \end{aligned} \end{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.


7.2


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,

\begin{equation} \dot y(t+h) = \dot y(t) - h y(t) \end{equation}
In [93]:
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
In [ ]:
 
In [94]:
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
In [95]:
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 $\dot y(0) = 0$:

In [73]:
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

In [130]:
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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