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Finite Difference Methods for Partial Differential Equations

  • Jakob Andreas BærentzenEmail author
  • Jens Gravesen
  • François Anton
  • Henrik Aanæs

Abstract

In this chapter, we initially give an introduction to methods for computing derivatives and partial derivatives using discrete differential operators and discuss the connection to Taylor series.

The chapter moves on to the topic of solving PDEs using finite difference methods. We discuss implicit and explicit methods and boundary conditions. The chapter also covers the categories of PDEs: elliptic, hyperbolic and parabolic as well as the important notions of consistence, convergence and stability. Finally, there is a discussion of 2D and 3D problems and problems on irregular grids.

At the end of the chapter, linear interpolation is discussed. We discuss both linear interpolation on a regular 2D or 3D grid and linear interpolation in a simplicial complex using barycentric coordinates.

As in the previous two chapters, this chapter is intended as a brush up and a point of reference. The reader who wishes to know more is referred to the literature.

Keywords

Finite Difference Method Finite Difference Scheme Implicit Method Finite Difference Equation Irregular Grid 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Smith, G.D.: Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3rd edn. Oxford University Press, Oxford (1985) zbMATHGoogle Scholar
  2. 2.
    Morton, K., Mayers, D.: Numerical Solution of Partial Differential Equations: An Introduction. Cambridge University Press, Cambridge (2005) zbMATHCrossRefGoogle Scholar
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    Osher, S.J., Fedkiw, R.P.: Level Set Methods and Dynamic Implicit Surfaces, 1st edn. Springer, Berlin (2002) Google Scholar
  4. 4.
    Leveque, R.J.: Numerical Methods for Conservation Laws. Birkhäuser, Boston (1992) zbMATHCrossRefGoogle Scholar
  5. 5.
    Moller, T., Machiraju, R., Mueller, K., Yagel, R.: A comparison of normal estimation schemes. In: Proceedings. Visualization’97 (Cat. No. 97CB36155), pp. 19–26, 525 (1997) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2012

Authors and Affiliations

  • Jakob Andreas Bærentzen
    • 1
    Email author
  • Jens Gravesen
    • 2
  • François Anton
    • 1
  • Henrik Aanæs
    • 1
  1. 1.Department of Informatics and Mathematical ModellingTechnical University of DenmarkKongens LyngbyDenmark
  2. 2.Department of MathematicsTechnical University of DenmarkKongens LyngbyDenmark

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