Second-Order Runge-Kutta Method (RK2)

Second-Order Runge-Kutta Method (RK2)#

This example demonstrates the second-order Runge-Kutta (RK2) method for solving ordinary differential equations (ODEs). The method is implemented to solve a first-order ODE of the form:

\[ \frac{dy}{dt} = f(t,y) = \sin^2(t) \cdot y \]

with initial condition \(y(0) = 2.0\) over the time interval \([0,5]\).

Method Description#

The RK2 method (also known as the midpoint method) uses two stages to compute each time step:

  1. First stage (slope at the beginning):

\[ k_1 = f(t_i, y_i) \]

  1. Second stage (slope at midpoint):

\[ k_2 = f(t_i + \frac{h}{2}, y_i + \frac{h}{2}k_1) \]

  1. Solution update:

\[ y_{i+1} = y_i + h \cdot k_2 \]

where:

  • \(h\) is the step size (set to 0.1 in the examples)

  • \(t_i\) is the current time

  • \(y_i\) is the solution at time \(t_i\)

This example is implemented in:

Both implementations include visualization of the solution using plotting tools (MATLAB’s built-in plot function and GNUplot for C++).