What is the 4th-order Runge-Kutta method for differential equations?

The 4th-order Runge-Kutta method is a numerical technique for approximating solutions to ordinary differential equations of the form y' = f(x, y) with initial value y(x₀) = K using weighted increments k₁, k₂, k₃, and k₄.

Differential Equations – Runge-Kutta Method Calculator & Tutorial

About the Runge-Kutta Method

This is an interactive tool to explore the numerical Runge-Kutta method. This method is used to approximate solutions to differential equations and is very powerful for solving a wide range of problems in science and engineering.

Consider the differential equation:

\( y' = f(x, y) \) with \( y(x_0) = K \) (initial value)

We wish to approximate the solution over an interval \( [a, b] \). We divide this interval into \( n \) smaller intervals of size \( h \). The 4th order Runge-Kutta method provides an approximation as follows:

Let \( y_0 = K \) (starting value)
\( y_{i+1} = y_i + \frac{1}{6}[k_1 + 2k_2 + 2k_3 + k_4] \) for \( i = 0, 1, \dots, n-1 \)

where:
\( k_1 = h \cdot f(x_i, y_i) \)
\( k_2 = h \cdot f(x_i + \frac{h}{2}, y_i + \frac{k_1}{2}) \)
\( k_3 = h \cdot f(x_i + \frac{h}{2}, y_i + \frac{k_2}{2}) \)
\( k_4 = h \cdot f(x_i + h, y_i + k_3) \)

The local truncation error is of the order \( O(h^5) \) and in principle decreases as \( h \) decreases.

Equations Used in This Demo

\( y' = x^2 \), \( y(0) = 1 \), exact solution: \( y = \frac{x^3}{3} + 1 \)
\( y' = x^4 \), \( y(0) = 1 \), exact solution: \( y = \frac{x^5}{5} + 1 \)
\( y' = e^x \), \( y(0) = 1 \), exact solution: \( y = e^x \)
\( y' = \frac{1}{1+x} \), \( y(0) = 1 \), exact solution: \( y = \ln(1+x) + 1 \)

Tutorial

All differential equations in this demo have the same initial value \( y(0) = 1 \) and exact solutions for comparison.

Select the first differential equation \( y' = x^2 \). At the start, \( h = 1.25 \) and \( n = 8 \).
Examine the exact (blue curve) and approximate (red points) solutions on the graph.
Decrease \( h \) by increasing \( n \) (or decrease \( h \) directly) and observe how the approximation improves.
Select the other differential equations and analyze the results.
Compare the exact and approximate values in the table below the graph.

Runge-Kutta Calculator

Differential Equation:

Step Size (h):

Number of Steps (n):

Results

Comparison between exact solution and Runge-Kutta approximation:

Step (i)	xᵢ	Approximate yᵢ	Exact y(xᵢ)	Error	k₁	k₂	k₃	k₄

Additional Resources

More references on differential equations: