Runge-Kutta method

A Runge-Kutta method is a type of numerical method of evaluating integrals and differential equations. The methods represent a generalization of what might be termed the straightforward method (Euler method). Whereas the Euler method effectively uses approximations of the values of a function based on the nearest known values along with the function's apparent slope indicated by those values, a Runge-Kutta method uses a weighted average of a set number of the functions' nearby values to work out a 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. For a particular use, such a tradeoff means there are choices to search for optimum calculation-methods that produce the required accuracy. The Runge-Kutta methods encompass a range of sophistication for this tradeoff, ranging from very simple steps, to steps of arbitrary complexity.