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I found that it is relatively easy to find a book that discusses Euler discretization or Runge-Kutta discretization, but I am not aware of one that is well-known and/or common knowledge (i.e., field-bible).

Can anyone provide a good reference in this area?

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  • $\begingroup$ You might find some more help at Computational Science; there are a few very active users who work in numerical methods for ODE there. $\endgroup$
    – Federico Poloni
    Commented Jul 29, 2022 at 7:13

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Too long for a comment: I think the list of references appearing on the wiki page for numerical methods for ordinary differential equations is not bad overall; let me just highlight some canonical references by (i) Ernst Hairer, Syvert Nørsett and Gerhard Wanner; (ii) Arieh Iserles; (iii) John Butcher; and (iv) C. William Gear.

(i) Hairer, Ernst; Nørsett, Syvert P.; Wanner, Gerhard, Solving ordinary differential equations. I: Nonstiff problems., Springer Series in Computational Mathematics 8. Berlin: Springer (ISBN 978-3-642-05163-0/pbk). xv, 528 p. (2010). ZBL1185.65115.

(ii) Iserles, Arieh, A first course in the numerical analysis of differential equations, Cambridge Texts in Applied Mathematics. Cambridge: Cambridge Univ. Press. 400 p. (1995). ZBL0841.65001.

(iii) Butcher, J. C., Numerical methods for ordinary differential equations., Chichester: Wiley (ISBN 0-471-96758-0/hbk). xiv, 425 p. (2003). ZBL1040.65057.

(iv) Gear, C. William, Numerical initial value problems in ordinary differential equations., Englewood Cliffs, NJ: Prentice-Hall. xvii, 253 p. (1971). ZBL1145.65316.

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  • $\begingroup$ PS: this answer is only focussed on references for numerical solution of initial value problems for ODEs, and in particular, the standard references for related topics like boundary value problems are not included. $\endgroup$
    – Nawaf Bou-Rabee
    Commented Jul 29, 2022 at 11:46

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