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Numerical Differentiation

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Basic Concepts in Computational Physics

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Abstract

Finite differences are the backbone of numerical differentiation. Based on the introduction of discrete grid-points and on the approximation of the function by its values at these grid-points, the forward, backward, and central differences are introduced. From these definitions the finite difference approximation of derivatives follows immediately together with the errors inherent to these approximations. An operator technique allows a more formal description and provides better insight. Finally, partial derivatives are discussed within this context.

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Notes

  1. 1.

    Please note that the symbols \(\varDelta \), \(\nabla \), and \(\delta \) in Eqs. (2.8) can also be regarded as operators acting on \(f_i\). For a basic introduction to the theory of linear operators see for instance [1, 2].

  2. 2.

    From now on, all symbols like \(D_c\), \(\varDelta \), etc. are interpreted as operators.

  3. 3.

    We note in passing that the shift operators form the discrete translational group, a very important group in theoretical physics. Let \(E(n) = E^n\) denote the shift by \(n \in {\mathbb {N}}\) grid-points. We then have

    $$\begin{aligned} E(n)E(m) = E(n+m), \end{aligned}$$
    (2.19a)
    $$\begin{aligned} E(0) = 1\!\!1, \end{aligned}$$
    (2.19b)

    and

    $$\begin{aligned} E(n)^{-1} = E(-n), \end{aligned}$$
    (2.19c)

    which are the properties required to form a group. Here \(1\!\!1\) denotes the unity element. Moreover, we have

    $$\begin{aligned} E(n) E(m) = E(m) E(n), \end{aligned}$$
    (2.19d)

    i.e. it is an Abelian group. The group of discrete translations is usually denoted by \(\mathbb {T}^d\).

References

  1. Weidmann, J.: Lineare Operatoren in Hilberträumen, vol. 1: Grundlagen. Springer, Berlin (2000)

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  2. Davies, E.B.: Linear Operators and their Spectra. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, UK (2007)

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  3. Lapidus, L., Pinder, G.F.: Numerical Solution of Partial Differential Equations. Wiley, New York (1982)

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  4. Gockenbach, M.S.: Understanding and Implementing the Finite Element Method. Cambridge University Press, Cambridge, UK (2006)

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  5. LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, UK (2002)

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Author information

Authors and Affiliations

  1. Faculty of Physics, University of Duisburg-Essen, Lotharstrasse 1-21, 47048, Duisburg, Germany

    Benjamin A. Stickler

  2. Institute of Theoretical and Computational Physics, Graz University of Technology, Graz, Austria

    Ewald Schachinger

Authors
  1. Benjamin A. Stickler
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  2. Ewald Schachinger
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Correspondence to Benjamin A. Stickler .

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Stickler, B.A., Schachinger, E. (2014). Numerical Differentiation. In: Basic Concepts in Computational Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-02435-6_2

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