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Unconditional stability of second-order ADI schemes applied to multi-dimensional diffusion equations with mixed derivative terms. (English) Zbl 1161.65073
The paper presents unconditional stability of seond-order alternating direction implicit (ADI) schemes applied to multi-dimensional diffusion equations with mixed derivative terms. The authors have investigated an ADI scheme proposed by I. J. D. Craig and A. D. Sneyd [Comput. Math. Appl. 16, No. 4, 341–350 (1988; Zbl 0654.65072)] and its subsequent modifications and an ADI scheme proposed by W. Hundsdorfer and J. Verwer [Numerical solution of time-dependent advection-diffusion-reaction equations. Springer Series in Computational Mathematics. 33. (2003; Zbl 1030.65100)]. The present ADI scheme is more general than the well known ADI scheme proposed by Douglass and Douglass and Rachford. The stability analysis has been carried out for the multi-dimensional diffusion equations and it has been found that the scheme is unconditionally stable. Numerical experiments have been carried out for the problems of 2 and 3 spatial dimensions with sufficient accuracy.
Reviewer’s remark: The authors may take note of the other valuable researches on the ADI method, e.g. Int. Commun. Heat Mass Transfer 23, 845–854 (1996); “The ADI method for heat conduction problems in orthogonal curvilinear coordinate system”, Heat Transfer Eng. 14, 64–72 (1994); “ADI methods for solution of the transient heat conduction problem in conical geometry”, Indian J. Technology 24, 36–39 (1986); “ADI method for solution of the transient heat conduction problem in spherical geometry”, Num. Meth. Thermal Probl. 11, 107 (1981), etc.

65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K10 Second-order parabolic equations
Full Text: DOI
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