Unconditional stability of second-order ADI schemes applied to multi-dimensional diffusion equations with mixed derivative terms.

*(English)*Zbl 1161.65073The 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.

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.

Reviewer: K. N. Shukla (Coimbatore)

##### MSC:

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

35K10 | Second-order parabolic equations |

##### Keywords:

initial boundary value problems; diffusion equitions; finite difference method; ADI splitting scheme; Neumann stability analysis
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\textit{K. J. in 't Hout} and \textit{B. D. Welfert}, Appl. Numer. Math. 59, No. 3--4, 677--692 (2009; Zbl 1161.65073)

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##### References:

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