Machine learningExplicit Multistage

Runge-Kutta Method

The Runge-Kutta Method is a family of explicit numerical techniques for solving ordinary differential equations (ODEs) developed independently by Carl Runge in 1895 and Martin Kutta in 1901. The fourth-order variant (RK4) is one of the most widely used algorithms in computational science and engineering for time-stepping problems.

Open in MethodMindSoonVideoSoon

Read the full method

Members only

Sign in with a free account to read this section.

Sign in

Sources

  1. Runge, C. (1895). Ueber die numerische Auflösung von Differentialgleichungen. Mathematische Annalen, 46(2), 167–178. DOI: 10.1007/BF01446807
  2. Kutta, M. W. (1901). Beitrag zur näherungsweisen Integration totaler Differentialgleichungen. Zeitschrift für Mathematik und Physik, 46, 435–453. link
  3. Butcher, J. C. (2008). Numerical Methods for Ordinary Differential Equations (2nd ed.). Wiley. DOI: 10.1002/9780470753767
ScholarGateRunge-Kutta Method (Runge-Kutta Method for Numerical Integration). Retrieved 2026-06-04 from https://scholargate.app/en/numerical-methods/runge-kutta-method