There are several issues with your bib entry. How to typeset Störmer may not be the most severe one.
You use the unicode-encoded form of the "en-dash" glyph: – between the words "Stormer" and "Verlet". Do use -- instead, unless you use a unicode-aware TeX engine (XeTeX or LuaTeX) to compile the document.
In order to keep BibTeX from lowercasing the names Störmer and Verlet in the title field, encase them in curly braces.
Abbreviating the journal's name to Acta Numer. seems quite unnecessary. Instead, do write Acta Numerica. If you feel ambitious, you could set up a string variable that encodes whether or not the journal name should be abbreviated. If your paper is intended for submission to an academic journal, the journal will likely have its own, "house" rules regarding journal name abbreviations. Don't make their job more difficult by providing possibly non-standard abbrevations.
Remove "doi: " from the doi field and make sure to use a bibliography style that knows how to handle the doi field. Aside: If you don't already do so, be sure to load the url package.
To separate the authors' names in the author field, use the keyword and, not commas.
I'd also use the authors' full given names in the author field, and leave it up to the bibliography style to determine whether full or abbreviated given names should be shown in the formatted bibliography.
Writing St{\"o}rmer instead of Störmer in the title field should not be necessary if you either employ a unicode-aware TeX engine or (in case you employ pdfLaTeX, which is not fully unicode-aware) load the inputenc package with the option utf8.
The fully modified bib entry should therefore look something like this:
@article{Hairer03,
author = "Ernst Hairer and Christian Lubich and Gerhard Wanner",
title = "Geometric numerical integration illustrated by the
{St{\"o}rmer--Verlet} method",
journal = "Acta Numerica",
volume = 12,
pages = "399--450",
year = 2003,
doi = "10.1017/S0962492902000144",
}
If you use a bibliography style which doesn't process the doi field, but if you somehow believe it's really necessary to show the DOI string, just change the field
doi = "10.1017/S0962492902000144",
to
note = "doi: \url{10.1017/S0962492902000144}",
The note field is always processed, by pretty much every bibliography style that's out there, somewhere.
(You did remember to load the url package, right?)
A full MWE (minimum working example):
\RequirePackage{filecontents}
\begin{filecontents}{mybib.bib}
@article{Hairer03,
author = "Ernst Hairer and Christian Lubich and Gerhard
Wanner",
title = "Geometric numerical integration illustrated by the
{St{\"o}rmer--Verlet} method",
journal = "Acta Numerica",
volume = 12,
pages = "399--450",
year = 2003,
doi = "10.1017/S0962492902000144",
}
\end{filecontents}
\documentclass{article}
\usepackage{natbib}
\bibliographystyle{plainnat} % any bib style that processes the 'doi' field
\usepackage{url} % to process the contents of the 'doi' field
\begin{document}
\nocite{*}
\bibliography{mybib}
\end{document}
St{\"{o}}rmer? The entire non-ascii letter has to be in braces. – daleif Feb 15 '18 at 12:04author = "{E. Hairer, C. Lubich and G. Wanner}"is wrong, it should beauthor = "E. Hairer and C. Lubich and G. Wanner"(no double braces and names separated withand, see https://tex.stackexchange.com/q/36396/35864) – moewe Feb 15 '18 at 12:05"...",{...}is just fine for bibtex entries. – daleif Feb 15 '18 at 12:09"..."author = "E. Hairer and C. Lubich and G. Wanner",is more 'consistent'. In fact I prefer{...}, but that really does not matter. – moewe Feb 15 '18 at 12:25