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The Pillars of Science Read chapter Read first chapter

2021 | OriginalPaper | Chapter

3. Our Model Problem (Our First Encounter with the Explicit Euler)

Author : Tobias Weinzierl

Published in: Principles of Parallel Scientific Computing

Publisher: Springer International Publishing

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Abstract

A simple N-body problem serves as demonstrator for the topics discussed in upcoming sections. We introduce an explicit Euler for this showcase problem informally, and thus obtain a baseline toy code to play with. The introduction allows us to discriminate numerical approximations from analytical solutions.

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Footnotes
1
The notation here uses an m in \(\frac{m}{s^2}\) as unit, while the m in \(F=mg\) denotes a parameter. From hereon, I work quite lazily with units and just skip them.
 
2
Einstein developed his theories, since the classic Newton model struggled to explain certain effects: The misfit of Newton’s theory with observations—Newton’s theory cannot quantiatively correctly predict how light “bends” around objects, though it was pointed out in 1784 (unpublished) and 1801 that it does bend to some degree—was well-known since the 1859s, but it took till 1916 until someone came up with a new theory that explains the effect. After that, experimentalists needed another three years plus a total solar eclipse to run further experiments which eventually validated Einstein’s theories. This is a great historic illustration of the ping-pong between the two classic pillars  of science.
 
3
The notation here can be quickly misunderstood, so you have to be careful: \(\partial _t\) and \(\text {d}_t\) are derivatives. dt is the time in-between two time steps.
 
4
Mathematicians like analytic functions. That’s something completely different, namely functions that can infinitely often be differentiated in \(\mathbb {C}\). We always speak of analytical solutions here. Our “analytical” highlights how the solution has been obtained, i.e. via exact, symbolic manipulation of the equations. The “analytic” in contrast is a property of a function.
 
Metadata
Title
Our Model Problem (Our First Encounter with the Explicit Euler)
Author
Tobias Weinzierl
Copyright Year
2021
Book
Principles of Parallel Scientific Computing

Print ISBN: 978-3-030-76193-6

Electronic ISBN: 978-3-030-76194-3

Copyright Year: 2021

DOI
https://doi.org/10.1007/978-3-030-76194-3_3

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