RK4

RK4 refers to the classical Runge-Kutta 4th-order method, an explicit single-step technique for solving initial value problems of ordinary differential equations dy/dt = f(t, y) with a given starting value y(t0) = y0. The method advances a solution from t_n to t_{n+1} = t_n + h by computing four intermediate slopes, or stages, and then forming a weighted average to update the solution.

The standard procedure uses the following stages:

- k1 = f(t_n, y_n)

- k2 = f(t_n + h/2, y_n + (h/2) k1)

- k3 = f(t_n + h/2, y_n + (h/2) k2)

- k4 = f(t_n + h, y_n + h k3)

The next value is then y_{n+1} = y_n + (h/6)(k1 + 2k2 + 2k3 + k4). This yields a fourth-order accurate approximation, with local truncation error on the order of h^5 and a global error of order h^4.

RK4 is explicit, requires four evaluations of f per step, and is widely used for its balance of accuracy and computational effort. It is generally stable for non-stiff problems, but like other explicit methods, it is not suitable for stiff systems without very small time steps. In practice, RK4 is often used with fixed step sizes for simplicity, or with adaptive schemes that estimate error to adjust h. It belongs to the wider family of Runge-Kutta methods and is commonly used as a reference point for comparing higher-order or adaptive integrators.