Runge-Kutta method

A Runge-Kutta method is a type of numerical method of evaluating integrals and differential equations. They represent a generalization of what might be termed the straight-forward method (Euler method), a generalization that encompasses some other such methods as well. Whereas the Euler method approximates a subsequent value of a function by assuming it is roughly where its current slope indicates, the Runge-Kutta method uses a weighted average of a set number of iterations working out the likely value. Such methods proceed calculating values in a series of small steps (e.g., if it is a function of time, in time-steps from some initial time for which the value is known), but there is a trade-off: using more smaller steps can produce better approximations, but using more sophisticated math (with more arithmetic) for each step can do so as well. Such a tradeoff leaves some room to search for optimum calculation-methods that produce the required accuracy. The Runge-Kutta methods offer a range of sophistication for this tradeoff, ranging from very simple steps, to steps of arbitrary complexity.