RKF45

RKF45, short for Runge-Kutta-Fehlberg 4(5), is an explicit method for solving initial-value problems for ordinary differential equations dy/dt = f(t, y). It belongs to the family of embedded Runge-Kutta methods, offering a fifth-order accurate estimate of y_{n+1} together with a fourth-order estimate of the same quantity. The two estimates yield a local truncation error estimate without extra function evaluations, enabling automatic adjustment of the integration step size as the computation proceeds.

In each step, the method evaluates f at several intermediate points to construct the two estimates (the 5th-order and 4th-order solutions) with a single set of function evaluations; the difference between the two estimates provides an estimate of the local error. The step size h is accepted or rejected based on this error relative to user-specified tolerances; if accepted, y_{n+1} is set to the 5th-order value and h is updated for the next step using a formula such as h_{n+1} = h_n * safety * (tol/err)^(1/5), where safety is a factor less than 1. If the error is too large, the step is rejected and retried with a smaller h.

History and usage: The RKF45 method was introduced by Erwin Fehlberg in 1968 as part of the embedded Runge-Kutta family. It is widely implemented in numerical libraries for non-stiff ordinary differential equations. While robust and efficient for many problems, RKF45 is explicit and not well-suited for stiff systems; modern software often uses other embedded pairs (for example Dormand–Prince 4(5) or Cash–Karp 4(5)) that may offer better accuracy or performance for specific classes of problems.