显式欧拉
import numpy as np
from scipy.integrate import odeint
def f(x,y):
return y-2*x/y
def f_ode(y,x):
return y-2*x/y
def Explicit_Euler(f,a,b,y0,h):
x_p = np.linspace(a,b,int(1/h)+1)
n = len(x_p)
value = np.zeros(n)
value[0] = y0
for i in range(1,n):
value[i] = value[i-1]+h*f(x_p[i-1],value[i-1]) #x,y位置改了一下
result=[i for j in odeint(f_ode,1,x_p) for i in j] #精确值,再转化为一维的
for i in range(n):
print('x={:.2f}时显式欧拉的误差为:{:.8f}'.format(x_p[i],abs(value[i]-result[i])))
Explicit_Euler(f,0,1,1,0.2)
实验截图:
改进欧拉
import numpy as np
from scipy.integrate import odeint
def f(x,y):
return y-2*x/y
def f_ode(y,x):
return y-2*x/y
def Imporve_Euler(f,a,b,y0,h):
x_p = np.linspace(a,b,int(1/h)+1)
n = len(x_p)
value = np.zeros(n)
value[0] = y0
for i in range(1,n):
T1 = value[i-1]+h*f(x_p[i-1],value[i-1])
T2 = value[i-1]+h*f(x_p[i],T1)
value[i] = (T1+T2)/2
result=[i for j in odeint(f_ode,1,x_p) for i in j] #精确值,再转化为一维的
for i in range(n):
print('x={:.2f}时显式欧拉的误差为:{:.8f}'.format(x_p[i],abs(value[i]-result[i])))
Imporve_Euler(f,0,1,1,0.2)
实验截图:
龙格库塔
import numpy as np
from scipy.integrate import odeint
def f(x,y):
return y-2*x/y
def f_ode(y,x):
return y-2*x/y
def Runge_kutta(f,a,b,y0,h):
x_p = np.linspace(a,b,int(1/h)+1)
n = len(x_p)
value = np.zeros(n)
value[0] = y0
for i in range(1,n):
k1 = f(x_p[i-1],value[i-1])
k2 = f(x_p[i-1]+h/2,value[i-1]+h/2*k1)
k3 = f(x_p[i-1]+h/2,value[i-1]+h/2*k2)
k4 = f(x_p[i-1]+h,value[i-1]+h*k3)
value[i] = value[i-1]+h/6*(k1+2*k2+2*k3+k4)
result=[i for j in odeint(f_ode,1,x_p) for i in j] #精确值,再转化为一维的
for i in range(n):
print('x={:.2f}时显式欧拉的误差为:{:.8f}'.format(x_p[i],abs(value[i]-result[i])))
Runge_kutta(f,0,1,1,0.2)
实验截图:
方法 | 显式欧拉误差 | 改进欧拉误差 | 龙格库塔误差 |
---|---|---|---|
x=0.2 | 0.01678407 | 0.00345073 | 0.00001336 |
x=0.4 | 0.03169259 | 0.00667151 | 0.00002618 |
x=0.6 | 0.04825549 | 0.01046424 | 0.00004181 |
x=0.8 | 0.06863307 | 0.01540958 | 0.00006254 |
x=1.0 | 0.09489743 | 0.02215388 | 0.00009113 |