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2021 | OriginalPaper | Chapter

13. Accuracy and Appropriateness of Numerical Schemes

Author : Tobias Weinzierl

Published in: Principles of Parallel Scientific Computing

Publisher: Springer International Publishing

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Abstract

For the accuracy and stability of numerical codes, we have to ensure that a discretisation is consistent and is stable. Both depend on the truncation error, while we distinguish zero- and A-stability. We end up with the notion of convergence according to the Lax Equivalence Theorem, and finally discuss how we can compute the convergence order experimentally.

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Footnotes
1
N here is a counter for the time steps. It has nothing to do with the N from “N-body simulations”.
 
2
I call \(\lambda \) the material parameter. Different to flow through a subsurface medium, e.g., our \(\lambda \) is not really a material. But the name material here highlights that it is not a parameter determined by yet another equation but something fixed.
 
3
Some maths books define Lipschitz-continuity “simply” as \(|F(s_1)-F(s_2)| \le C |s_1-s_2|\). In our discussion, we split up this s into \(s=(t,f(t))\) as we are interested in ODEs, and we wobble around with the f(t) part only. The more general definition from math books shows that we also can slightly alter the t argument. The solution will not change too much either. Both definitions focus on the right-hand side of the ODE. As the right side determines the solution, its (continuity) properties carry over to the solution.
 
Metadata
Title
Accuracy and Appropriateness of Numerical Schemes
Author
Tobias Weinzierl
Copyright Year
2021
DOI
https://doi.org/10.1007/978-3-030-76194-3_13

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