Numerical methods for ordinary differential equations are computational schemes to obtain approximate solutions of ordinary differential equations (ODEs).

Background
Since ODEs appeared in science, many mathematicians have studied how to solve them. However, only few of them can be mathematically solved. This is why numerical methods are needed. One of the most famous methods are the Runge-Kutta methods, but it doesn't work for some ODEs (especially nonlinear ODEs). This is why new ODE solvers are developed. The following list includes frequently used methods:

 Bulirsch-Stoer algorithm
 Euler's method (named after Leonhard Euler) and their variants
 Backward Euler method
 Semi-implicit Euler method
 Euler-Maruyama method
 Exponential integrator
 Leapfrog method
 Linear multistep methods
 Shooting method
 Symplectic integrator
 Taylor series method

Validated Numerics for ODEs

Not only approximate solvers, but the study to "verify the existence of solution by computers" is also active. This study is needed because numerically obtained solutions could be phantom solutions (fake solutions). This kind of incident is already reported. The popular methods are based on the shooting method or spectral methods. Today, European research teams and Japanese experts are working on this topic.

ODEs and Related Topics Studied in the Context of Validated Numerics

 Blow-up solutions
 Lorentz equation
 Rossler equation

Related Software

Chebfun
COSY INFINITY 
 INTLAB and kv are interval arithmetic libraries which include ODE solvers
 MATLAB - made by MathWorks
 NAG library
 Wolfram Mathematica - made by Wolfram Research

References

Further reading

 Mitsui, T., & Shinohara, Y. (1995). Numerical analysis of ordinary differential equations and its applications. World Scientific.
 Iserles, A. (2009). A first course in the numerical analysis of differential equations. Cambridge University Press.
 Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems, Berlin, New York: Springer-Verlag.
 Wanner, G. & Hairer, E. (1996), Solving ordinary differential equations II: Stiff and differential-algebraic problems (2nd ed.). Springer Berlin Heidelberg.
 Butcher, John C. (2008), Numerical Methods for Ordinary Differential Equations, New York: John Wiley & Sons.
 John D. Lambert, Numerical Methods for Ordinary Differential Systems, John Wiley & Sons, Chichester, 1991.
 Deuflhard, P., & Bornemann, F. (2012). Scientific computing with ordinary differential equations. Springer Science & Business Media.
 Shampine, L. F. (2018). Numerical solution of ordinary differential equations. Routledge.
 Dormand, John R. (1996), Numerical Methods for Differential Equations: A Computational Approach, Boca Raton: CRC Press.

Other websites
 Joseph W. Rudmin, Application of the Parker–Sochacki Method to Celestial Mechanics , 1998.
 Dominique Tournès, L'intégration approchée des équations différentielles ordinaires (1671-1914), thèse de doctorat de l'université Paris 7 - Denis Diderot, juin 1996. Réimp. Villeneuve d'Ascq : Presses universitaires du Septentrion, 1997, 468 p. (Extensive online material on ODE numerical analysis history, for English-language material on the history of ODE numerical analysis, see e.g. the paper books by Chabert and Goldstine quoted by him.)
INTLAB
Verified ODE (IVP) Solver

Differential equations
Technology
Computer science
Computing