×

zbMATH — the first resource for mathematics

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.

MSC:
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K10 Second-order parabolic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Brian, P.L.T., A finite-difference method of high-order accuracy for the solution of three-dimensional transient heat conduction problems, Aiche j., 7, 367-370, (1961)
[2] Craig, I.J.D.; Sneyd, A.D., An alternating-direction implicit scheme for parabolic equations with mixed derivatives, Comp. math. appl., 16, 341-350, (1988) · Zbl 0654.65072
[3] Douglas, J., Alternating direction methods for three space variables, Numer. math., 4, 41-63, (1962) · Zbl 0104.35001
[4] Douglas, J.; Rachford, H.H., On the numerical solution of heat conduction problems in two and three space variables, Trans. amer. math. soc., 82, 421-439, (1956) · Zbl 0070.35401
[5] Duffy, D.J., Finite difference methods in financial engineering: A partial differential equation approach, (2006), Wiley · Zbl 1141.91002
[6] in ’t Hout, K.J.; Welfert, B.D., Stability of ADI schemes applied to convection – diffusion equations with mixed derivative terms, Appl. numer. math., 57, 19-35, (2007) · Zbl 1175.65104
[7] Hundsdorfer, W., Accuracy and stability of splitting with stabilizing corrections, Appl. numer. math., 42, 213-233, (2002) · Zbl 1004.65095
[8] Hundsdorfer, W.; Verwer, J.G., Numerical solution of time-dependent advection – diffusion – reaction equations, Springer series in computational mathematics, vol. 33, (2003), Springer-Verlag Berlin · Zbl 1030.65100
[9] Peaceman, D.W., Fundamentals of numerical reservoir simulation, developments in petroleum science, vol. 6, (1977), Elsevier Amsterdam
[10] Tavella, D.; Randall, C., Pricing financial instruments: the finite difference method, (2000), Wiley New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.