Runge-Kutta methods and Butcher tableau
The Runge-Kutta fourth-order method is a numerical method used to solve ordinary differential equations (ODEs) numerically. It provides a more accurate approximation of the solution compared to lower-order methods like Euler's method.
The Runge-Kutta fourth-order method is based on the idea of approximating the solution of an ODE by considering the slope at different points within a small interval. By calculating the weighted average of these slopes, the method provides a more accurate estimation of the next point on the solution curve.
The algorithm for the Runge-Kutta fourth-order method can be summarized as follows:
Given an initial value problem (IVP) of the form $dy/dx = f(x,y)$ with initial condition $y(x_0) = y_0$, where f(x, y) represents the derivative of y with respect to x.
Select a step size $h$.
Start with the initial values $x_0$ and $y_0$.
Repeat the following steps until the desired endpoint x is reached: a. Calculate the slopes $k_1$, $k_2$, $k_3$, and $k_4$ using the following formulas:
$$ \begin{e $$
k1 = h * f(xn, yn)
k2 = h * f(xn + h/2, yn + k1/2)
k3 = h * f(xn + h/2, yn + k2/2)
k4 = h * f(xn + h, yn + k3)
Update the values of x and y using the following formulas:
xn+1 = xn + h
yn+1 = yn + (k1 + 2k2 + 2k3 + k4)/6