Runge-Kutta method

It is a family of numerical methods to solve ODEs. The more famous is the RK4 (4th order).

\begin{array}{l} y_{n + 1} = y_{n} + h \frac{1}{6} (k_{1} + 2 k_{2} + 2 k_{3} + k_{4}) \\ t_{n + 1} = t_{n} + h \end{array}

where

\begin{array}{l} k_{1} = f (t_{n}, y_{n}) \\ k_{2} = f (t_{n} + \frac{h}{2}, y_{n} + h \frac{k_{1}}{2}), \\ k_{3} = f (t_{n} + \frac{h}{2}, y_{n} + h \frac{k_{2}}{2}), \\ k_{4} = f (t_{n} + h, y_{n} + h k_{3}) . \end{array}

Here $y_{n + 1}$ is the RK4 approximation of $y (t_{n + 1})$ , and the next value ( $y_{n + 1}$ ) is determined by the present value ( $y_{n}$ ) plus the weighted average of four increments, where each increment is the product of the size of the interval, $h$ , and an estimated slope specified by function f on the right-hand side of the differential equation.

Interpretation

Short answer: you can think of $v := \frac{1}{6} (k_{1} + 2 k_{2} + 2 k_{3} + k_{4})$ as a weighted average slope, so

y_{n + 1} = y_{n} + h v_{n}

is nothing but a kind of Euler method but with an improved slope. Instead of taking the slope at $(x_{n}, y_{n})$ you take several measures ( $k_{i}$ ) in the interval $(x_{n}, x_{n + 1})$ . Why these weights exactly? This requires a long answer.

Long answer: see this great answer. To summarize, the exact value of $y_{n + 1}$ is $y_{n} + \int_{x_{n}}^{x_{n + 1}} f (x, y) d x$ so you should approximate this integral numerically (because you don't know $y (x)$ ). When you use Simpson's rule you obtain the formula you have presented.