Diffusion Problem

Røed, Lars Petter

doi:10.1007/978-3-319-93864-6_4

Diffusion Problem

Lars Petter Røed^2,3

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First Online: 05 September 2018

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Part of the book series: Springer Textbooks in Earth Sciences, Geography and Environment ((STEGE))

Abstract

In this chapter, we discuss the fundamentals of how to cast a PDE into finite difference form. More specifically, the reader will learn how to discretize the diffusion equation, and learn why some discretizations work and some do not. This will be the opportunity to introduce concepts such as numerical stability, convergence, and consistency. It will be explained how to check whether a discretization is stable and consistent, and the reader will learn about explicit and implicit schemes, the rudiments of elliptic solvers, and the concept of numerical dissipation or artificial damping inherent in our discretizations.

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Notes

1.
Only wavenumbers for which \(\lambda =0\) have a growth factor \(|G|=1\). However, all wavenumbers are present in any solution. Hence, the growth factor is in general less than unity.
2.
It is not important how fast each of them go to zero.
3.
For a given grid size, \(2\Delta x\) is the so-called Nüquist wavelength. Thus, wavelengths smaller than \(2\Delta x\) remain unresolved.
4.
As a corollary, we note that this proves that the truly implicit scheme (4.64), which follows from (4.65) by letting \(\gamma =1\), is indeed unconditionally stable.
5.
This is also commonly referred to as the mesh or grid size.
6.
These are oscillations in which the frequency equals the inertial frequency f.

References

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Authors and Affiliations

Department of Geosciences, University of Oslo, Oslo, Norway
Lars Petter Røed
Norwegian Meteorological Institute, Oslo, Norway
Lars Petter Røed

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Lars Petter Røed
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Correspondence to Lars Petter Røed .

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Røed, L.P. (2019). Diffusion Problem. In: Atmospheres and Oceans on Computers. Springer Textbooks in Earth Sciences, Geography and Environment. Springer, Cham. https://doi.org/10.1007/978-3-319-93864-6_4

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DOI: https://doi.org/10.1007/978-3-319-93864-6_4
Published: 05 September 2018
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-93863-9
Online ISBN: 978-3-319-93864-6
eBook Packages: Earth and Environmental ScienceEarth and Environmental Science (R0)

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