We use Runge-Kutta methods as a class to represent what are also called single-step or one-step methods. A gentle introduction to these and other kinds of IVP methods can be found in Atkinson and Han Atkinson (1989). More advanced introductions are given in Corless and Fillon Corless & Fillion (2013) and Iserles Iserles (1996). The most definitive reference is by Hairer et al. Hairer et al. (2008).
A dated but still interesting article about the built-in functions for solving initial value problems in MATLAB is by Shampine Shampine & Reichelt (1997). Methods for a more general type of problem known as differential–algebraic equations are covered in Brenan et al. Brenan et al. (1996).
Interesting history of IVP methods can be found at the SIAM website, where C. W. Gear gives both an oral history and an article reprinted from Nash (1990).
- Atkinson, K. E. (1989). An Introduction to Numerical Analysis (2nd ed). Wiley.
- Corless, R. M., & Fillion, N. (2013). A Graduate Introduction to Numerical Methods: From the Viewpoint of Backward Error Analysis. Springer Science & Business Media.
- Iserles, A. (1996). A First Course in the Numerical Analysis of Differential Equations. Cambridge University Press.
- Hairer, E., Nørsett, S. P., & Wanner, G. (2008). Solving Ordinary Differential Equations I: Nonstiff Problems. Springer Science & Business Media.
- Shampine, L. F., & Reichelt, M. W. (1997). The MATLAB ODE Suite. SIAM Journal on Scientific Computing, 18(1), 1–22. 10.1137/S1064827594276424