Runge-Kutta 2nd order method to solve Differential equations //

Given the following inputs:

An ordinary differential equation that defines the value of dy/dx in the form x and y.
$\LARGE \frac{dy}{dx} = f(x, y)$
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 the next value y_n+1 from the previous value y_n. Therefore:

y_n+1 = value of y at (x = n + 1)
y_n = value of y at (x = n)
where
  0 ? n ? (x - x₀)/h
  h is step height
  x_n+1 = x₀ + h

The essential formula to compute the value of y(n+1):

K1 = h * f(x,y)
K2 = h * f(x/2, y/2) or K1/2
y_{n+1 =}y_n+K²+ (h³)

The formula basically computes the next value y_n+1 using current y_n 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 in the order of O(h³), while the total accumulated error is order of O(h⁴).

Below is the implementation of the above approach:

C++

// C++ program to implement Runge
// Kutta method
 
#include <bits/stdc++.h>
 
using namespace std;
 
// 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;
 
    cout << fixed << setprecision(6)
         << "y(x) = " << rungeKutta(x0, y, x, h);
    return 0;
}
 
// This code is contributed by shivani

Java

// 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

Python3

# 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

C

// 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;
}

C#

// 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

Javascript

<script>
 
// JavaScript program to implement Runge
// Kutta method
 
    // A sample differential equation
    // "dy/dx = (x - y)/2"
    function 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.
    function rungeKutta(x0, y0, x, h)
    {
        // Count number of iterations
        // using step size or
        // step height h
        let n = ((x - x0) / h);
     
        let k1, k2;
     
        // Iterate for number of iterations
        let y = y0;
        for (let 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
     
    let x0 = 0, y = 1,
              x = 2, h = 0.2;
     
    document.write(rungeKutta(x0, y, x, h).toFixed(6));
   
                   
</script>