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ADI Methods

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Encyclopedia of Applied and Computational Mathematics

Mathematics Subject Classification

65M06; 65N06

Synonyms

Alternating direction implicit method; Splitting method

Description

Finite-difference methods have been extremely important to the numerical solution of partial differential equations. An ADI method is one of them with extraordinary features in structure simplicity, computational efficiency, and flexibility in applications.

The original ADI idea was proposed by D. W. Peaceman and H. H. Rachford, Jr., [12] in 1955. Later, J. Douglas, Jr., and H. H. Rachford, Jr., [3] were able to implement the algorithm by splitting the time-step procedure into two fractional steps. The strategy of the ADI approach can be readily explained in a contemporary way of modern numerical analysis. To this end, we let \(\mathcal{D}\) be a two-dimensional spacial domain and consider the following partial differential equation:

$$\displaystyle{ \frac{\partial u} {\partial t} = \mathcal{F}u + \mathcal{G}u,\ \ \ (x,y) \in \mathcal{D},\ t> t_{0}, }$$
...

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Author information

Authors and Affiliations

  1. Department of Mathematics, Baylor University, One Bear Place, 76798-7328, Waco, TX, USA

    Qin Sheng

Authors
  1. Qin Sheng
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Editor information

Editors and Affiliations

  1. University of Texas at Austin, Austin, TX, USA

    Björn Engquist

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Sheng, Q. (2015). ADI Methods. In: Engquist, B. (eds) Encyclopedia of Applied and Computational Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70529-1_435

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