On the stability of ADI methods

  • Ole Østerby DEPARTMENT OF COMPUTER SCIENCE AARHUS UNIVERSITY IT-parken, Aabogade 34 DK-8200 Aarhus N, Denmark


When solving parabolic equations in two space dimensions implicit methods are preferred to the explicit method because of their better stability properties. Straightforward implementation of implicit methods require time-consuming solution of large systems of linear equations, and ADI methods are preferred instead. We expect the ADI methods to inherit the stability properties of the implicit methods they are derived from, and we demonstrate that this is partly true. The Douglas-Rachford and Peaceman-Rachford methods are absolutely stable in the sense that their growth factors are ≤ 1 in absolute value. Near jump discontinuities, however, there are differences w.r.t. how the ADI methods react to the situation: do they produce oscillations and how effectively do they damp them. We demonstrate the behaviour on two simple examples.


G. G. O’Brien, M. A. Hyman, and S. Kaplan,

A Study of the Numerical Solution of Partial Differential Equations,

J. Math. Phys., 29 (1951), pp. 223–251. doi:10.1002/sapm1950291223

J. Crank and P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type,

Proc. Cambridge Philos. Soc., 43 (1947), pp. 50–67.

Reprinted in Adv. Comput. Math., 6 (1996), pp. 207–226.


J. Douglas and H. H. Rachford, On the numerical solution of heat conduction problems in two and three space variables,

Trans. Amer. Math. Soc., 82 (1956), pp. 421–439.


P. M. Gresho and R. L. Lee, Don’t suppress the wiggles – they’re telling you something, Computers and Fluids, 9 (1981), pp. 223–253.


P. Laasonen, Über eine Methode zur Lösung der Wärmeleitungsgleichung,

Acta Math., 81 (1949), pp. 309–317.

D. W. Peaceman and H. H. Rachford, The numerical solution of parabolic and elliptic differential equations, J. SIAM, 3 (1955), pp. 28–41.


O. Østerby, Five ways of reducing the Crank-Nicolson oscillations,

BIT, 43 (2003), pp. 811–822. doi:10.1023/B:BITN.0000009942.00540.94

O. Østerby, Numerical Solution of Parabolic Equations,

Department of Computer Science, Aarhus University, 2015


How to Cite
Østerby, O. (2016). On the stability of ADI methods. DAIMI Report Series, 42(598). https://doi.org/10.7146/dpb.v42i598.25146