Numerically solving initial value problem

(공학수학1 과제용)

Let's solve an initial value problem numerically with a computer.

We will compare the accuracy of three methods: Euler method, Improved Euler method, and Runge-Kutta method.

Problem

$y'=y+x^2$ , $y(0) = 0$

Source Code (Python)

x0 = 0.0
y0 = 0.0
h = 0.1
exact = 0.436563656918

def f(x,y):
    return y + x**2

# (1) Euler method
x = x0
y = y0
for i in range(10):
    Dy = f(x, y)
    x += h
    y += Dy*h

print("(1) Euler method")    
print("y =", y)
print("e =", abs(y-exact))

# (2) Improved Euler method
x = x0
y = y0
for i in range(10):
    Dy = f(x, y)
    Dy2 = f(x + h, y + Dy*h)
    x += h
    y += (h/2) * (Dy + Dy2)  

print("\n(2) Improved Euler method")    
print("y =", y)
print("e =", abs(y-exact))


# (3) Runge-Kutta method
x = x0
y = y0
for i in range(10):
    k1 = h*f(x,y)
    k2 = h*f(x + h/2, y + k1/2)
    k3 = h*f(x + h/2, y + k2/2)
    k4 = h*f(x+h, y+k3)
    x += h
    y += (k1 + 2*k2 + 2*k3 + k4) / 6

print("\n(3) Runge-Kutta method")    
print("y =", y)
print("e =", abs(y-exact))

Results

(1) Euler method
y = 0.34685916621
e = 0.08970449070799996


(2) Improved Euler method
y = 0.4363239829622023
e = 0.00023967395579765904


(3) Runge-Kutta method
y = 0.43656289201769494
e = 7.649003050391734e-07