Runge-Kutta 2nd order method to solve Differential equations

Given the following inputs:

  1. An ordinary differential equation that defines the value of dy/dx in the form x and y.

    \LARGE \frac{dy}{dx} = f(x, y)

  2. Initial value of y, i.e., y(0).

    \LARGE y(0)= y_o

The task is to find the value of unknown function y at a given point x, i.e. y(x).

Example:

Input: x0 = 0, y0 = 1, x = 2, h = 0.2
Output: y(x) = 0.645590

Input: x0 = 2, y0 = 1, x = 4, h = 0.4;
Output: y(x) = 4.122991



Approach:
The Runge-Kutta method finds an approximate value of y for a given x. Only first-order ordinary differential equations can be solved by using the Runge Kutta 2nd order method.

Below is the formula used to compute next value yn+1 from previous value yn.

Therefore:

yn+1 = value of y at (x = n + 1)
yn = value of y at (x = n)
where
  0 ≤ n ≤ (x - x0)/h
  h is step height
  xn+1 = x0 + h

The essential formula to compute the value of y(n+1):
\LARGE K_{1} = h*f(x_{n}, y_{n})
\LARGE K_{2} = h*f((x_{n} + \frac{h}{2}), (y_{n} + \frac{K_{1}*h}{2}))
\LARGE y_{n+1} = y_{n} + K_{2} + (h^{3})

The formula basically computes the next value yn+1 using current yn plus the weighted average of two increments:

  • K1 is the increment based on the slope at the beginning of the interval, using y.
  • K2 is the increment based on the slope at the midpoint of the interval, using (y + h*K1/2).

The method is a second-order method, meaning that the local truncation error is on the order of O(h3), while the total accumulated error is order O(h4).

Below is the implementation of the above approach:

C++

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// C program to implement Runge
// Kutta method
  
#include <stdio.h>
  
// A sample differential equation
// "dy/dx = (x - y)/2"
float dydx(float x, float y)
{
    return (x + y - 2);
}
  
// Finds value of y for a given x
// using step size h
// and initial value y0 at x0.
float rungeKutta(float x0, float y0,
                 float x, float h)
{
    // Count number of iterations
    // using step size or
    // step height h
    int n = (int)((x - x0) / h);
  
    float k1, k2;
  
    // Iterate for number of iterations
    float y = y0;
    for (int i = 1; i <= n; i++) {
        // Apply Runge Kutta Formulas
        // to find next value of y
        k1 = h * dydx(x0, y);
        k2 = h * dydx(x0 + 0.5 * h,
                      y + 0.5 * k1);
  
        // Update next value of y
        y = y + (1.0 / 6.0) * (k1 + 2 * k2);
  
        // Update next value of x
        x0 = x0 + h;
    }
  
    return y;
}
  
// Driver Code
int main()
{
    float x0 = 0, y = 1,
          x = 2, h = 0.2;
  
    printf("y(x) = %f",
           rungeKutta(x0, y, x, h));
    return 0;
}

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Java

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// Java program to implement Runge
// Kutta method
class GFG {
      
    // A sample differential equation
    // "dy/dx = (x - y)/2"
    static double dydx(double x, double y)
    {
        return (x + y - 2);
    }
      
    // Finds value of y for a given x
    // using step size h
    // and initial value y0 at x0.
    static double rungeKutta(double x0, double y0,
                     double x, double h)
    {
        // Count number of iterations
        // using step size or
        // step height h
        int n = (int)((x - x0) / h);
      
        double k1, k2;
      
        // Iterate for number of iterations
        double y = y0;
        for (int i = 1; i <= n; i++) {
            // Apply Runge Kutta Formulas
            // to find next value of y
            k1 = h * dydx(x0, y);
            k2 = h * dydx(x0 + 0.5 * h,
                          y + 0.5 * k1);
      
            // Update next value of y
            y = y + (1.0 / 6.0) * (k1 + 2 * k2);
      
            // Update next value of x
            x0 = x0 + h;
        }
      
        return y;
    }
      
    // Driver Code
    public static void main (String[] args)
    {
        double x0 = 0, y = 1,
              x = 2, h = 0.2;
      
        System.out.println(rungeKutta(x0, y, x, h));
    }
}
  
// This code is contributed by Yash_R

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Python3

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# Python3 program to implement Runge 
# Kutta method 
  
# A sample differential equation 
# "dy/dx = (x - y)/2" 
def dydx(x, y) :
  
    return (x + y - 2); 
  
# Finds value of y for a given x 
# using step size h 
# and initial value y0 at x0. 
def rungeKutta(x0, y0, x, h) : 
  
    # Count number of iterations 
    # using step size or 
    # step height h 
    n = round((x - x0) / h);
      
        # Iterate for number of iterations 
    y = y0; 
      
    for i in range(1, n + 1) :
          
                # Apply Runge Kutta Formulas 
        # to find next value of y 
        k1 = h * dydx(x0, y); 
        k2 = h * dydx(x0 + 0.5 * h, y + 0.5 * k1); 
  
        # Update next value of y 
        y = y + (1.0 / 6.0) * (k1 + 2 * k2); 
  
        # Update next value of x 
        x0 = x0 + h; 
  
    return y; 
  
# Driver Code 
if __name__ == "__main__"
  
    x0 = 0; y = 1
    x = 2; h = 0.2
  
    print("y(x) =",rungeKutta(x0, y, x, h)); 
  
# This code is contributed by Yash_R

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C#

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// C# program to implement Runge
// Kutta method
using System;
  
class GFG {
      
    // A sample differential equation
    // "dy/dx = (x - y)/2"
    static double dydx(double x, double y)
    {
        return (x + y - 2);
    }
      
    // Finds value of y for a given x
    // using step size h
    // and initial value y0 at x0.
    static double rungeKutta(double x0, double y0,
                     double x, double h)
    {
        // Count number of iterations
        // using step size or
        // step height h
        int n = (int)((x - x0) / h);
      
        double k1, k2;
      
        // Iterate for number of iterations
        double y = y0;
        for (int i = 1; i <= n; i++) {
            // Apply Runge Kutta Formulas
            // to find next value of y
            k1 = h * dydx(x0, y);
            k2 = h * dydx(x0 + 0.5 * h,
                          y + 0.5 * k1);
      
            // Update next value of y
            y = y + (1.0 / 6.0) * (k1 + 2 * k2);
      
            // Update next value of x
            x0 = x0 + h;
        }
      
        return y;
    }
      
    // Driver Code
    public static void Main (string[] args)
    {
        double x0 = 0, y = 1,
              x = 2, h = 0.2;
      
        Console.WriteLine(rungeKutta(x0, y, x, h));
    }
}
  
// This code is contributed by Yash_R

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Output:

y(x) = 0.645590

Reference: https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods

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