Numerical methods for ordinary differential equations

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; Norsett, 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 Tournes, L'integration approchee des equations differentielles ordinaires (1671-1914), these de doctorat de l'universite Paris 7 - Denis Diderot, juin 1996. Reimp. 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
Numerical analysis