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:
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.