Bottom SciencePython Programming
Basics
Numerical Methods
- Bisection Method
- Secant Method
- Regular Falsi (False Position) Method
- Newton Raphson Method
- Gauss Elimination Method
- Gauss Jordan Method
- Gauss-Seidel Method
- Lagrange Interpolation Method
- Newton Divided Difference Interpolation
- Newton Forward Difference Interpolation
- Newton Backward Difference Interpolation
- Trapezoidal Rule
- Simpson 1/3rd Rule
- Simpson 3/8 Rule
- Euler’s Method
- Euler’s Modified Method
- Runge-Kutta 2nd Order Method
- Runge-Kutta 4th Order Method
- Cubic Spline Method
- Bilinear Interpolation Method
- Milne’s Method
- More topics coming soon…
Advance Numerical Methods
Euler’s modified method, also known as the improved Euler method, is a variation of Euler’s method that improves the accuracy of the solution.
It is a first-order method, but it uses the slope at the midpoint of the interval, rather than the slope at the starting point to estimate the value of the solution at the next time step.
Here is a simple example of how to implement Euler’s modified method in Python for solving the ODE dy/dx = -y with the initial value y(0) = 1:
Euler’s Modified Method in Python
import numpy as np
def euler_modified_method(f, y0, x0, x_end, h):
# Initialize the solution array
x = np.arange(x0, x_end+h, h)
y = np.zeros(len(x))
y[0] = y0
# Euler's Modified Method [By Bottom Science]
# Iterate over the steps
for i in range(1, len(x)):
k1 = h*f(x[i-1], y[i-1])
k2 = h*f(x[i-1] + h/2, y[i-1] + k1/2)
y[i] = y[i-1] + k2
return x, y
# Define the ODE function
def f(x, y):
return -y
# Set the initial condition and the step size
y0 = 1
x0 = 0
x_end = 10
h = 0.1
# Solve the ODE
x, y = euler_modified_method(f, y0, x0, x_end, h)
for xx,yy in zip(x,y):
print("x = ",xx," y = ",yy)
