In numerical analysis, the Runge–Kutta methods (English: /ˈrʊŋəˈkʊtɑː/ ⓘ RUUNG-ə-KUUT-tah[1]) are a family of implicit and explicit iterative methods, which include the Euler method, used in temporal discretization for the approximate solutions of simultaneous nonlinear equations.[2] These methods were developed around 1900 by the German mathematicians Carl Runge and Wilhelm Kutta. Slopes used by