zbMATH — the first resource for mathematics

A high-order Padé ADI method for unsteady convection-diffusion equations. (English) Zbl 1089.65092
Summary: A high-order alternating direction implicit (ADI) method for computations of unsteady convection-diffusion equations is proposed. By using fourth-order Padé schemes for the spatial derivatives, the present scheme is fourth-order accurate in space and second-order accurate in time. The solution procedure consists of a number of tridiagonal matrix operations which make the computation cost effective. The method is unconditionally stable, and shows higher accuracy and better phase and amplitude error characteristics than the standard second-order ADI method by D. W. Peaceman and H. H. Rachford jun. [J. Soc. Ind. Appl. Math. 3, 28–41 (1955; Zbl 0067.35801)] and the fourth-order ADI scheme of S. Karaa and J. Zhang [J. Comput. Phys. 198, 1–9 (2004; Zbl 1053.65067)].

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
Full Text: DOI
[1] Peaceman, D.W.; Rachford, H.H., The numerical solution of parabolic and elliptic differential equations, Journal of the society of industrial and applied mathematics, 3, 28-41, (1959) · Zbl 0067.35801
[2] Karaa, S.; Zhang, J., High order ADI method for solving unsteady convection-diffusion problems, Journal of computational physics, 198, 1-9, (2004) · Zbl 1053.65067
[3] Kalita, J.C.; Dalal, D.C.; Dass, A.K., A class of higher order compact schemes for the unsteady two-dimensional convection-diffusion equation with variable convection coefficients, International journal for numerical methods in fluids, 38, 1111-1131, (2002) · Zbl 1094.76546
[4] Spotz, W.F.; Carey, G.F., Extension of high-order compact schemes to time-dependent problems, Numerical methods for partial differential equations, 17, 657-672, (2001) · Zbl 0998.65101
[5] Rigal, A., High order difference schemes for unsteady one-dimensional diffusion-convection problems, Journal of computational physics, 114, 59-76, (1994) · Zbl 0807.65096
[6] Noye, B.J.; Tan, H.H., Finite difference methods for solving the two-dimensional advection-diffusion equation, International journal for numerical methods in fluids, 26, 1615-1629, (1988) · Zbl 0638.76104
[7] van der Houwen, P.J.; de Vries, H.B., Fourth order ADI method for semidiscrete parabolic equations, Journal of computational and applied mathematics, 9, 1, 41-63, (1983) · Zbl 0513.65051
[8] de Vries, H.B., Comparative study of ADI splitting methods for parabolic equations in two space dimensions, Journal of computational and applied mathematics, 10, 2, 179-193, (1984) · Zbl 0574.65132
[9] van der Houwen, P.J., Iterated splitting methods of high order for time-dependent partial differential equations, SIAM journal on numerical analysis, 21, 4, 635-656, (1984) · Zbl 0566.65054
[10] MacKinnon, R.J.; Carey, G.F., Analysis of material interface discontinuities and superconvergent fluxes in finite difference theory, Journal of computational physics, 75, 151-167, (1988) · Zbl 0632.76110
[11] H. Choi, P. Moin, J. Kim, Turbulent drag reduction: studies of feedback control and flow over riblets, Report TF-55, Department of Mechanical Engineering, Stanford University, Stanford, California, September 1992.
[12] K. Akselvoll, P. Moin, Large eddy simulation of turbulent confined coannular jets and turbulent flow over a backward facing step, Report TF-63, Department of Mechanical Engineering, Stanford University, Stanford, California, February 1995.
[13] You, D.; Mittal, R.; Wang, M.; Moin, P., Computational methodology for large-eddy simulation of tip-clearance flows, AIAA journal, 42, 2, 271-279, (2004)
[14] Visbal, M.R.; Gaitonde, D.V., High-order-accurate methods for complex unsteady subsonic flows, AIAA journal, 37, 10, 1231-1239, (1999)
[15] Mittal, R.; Moin, P., Suitability of upwind-biased schemes for large-eddy simulation of turbulent flows, AIAA journal, 36, 1415-1417, (1997) · Zbl 0900.76336
[16] Moin, P., Fundamentals of engineering numerical analysis, (2001), Cambridge University Press · Zbl 0993.65003
[17] Hundsdorfer, W.H.; Verwer, J.G., Stability and convergence of the peaceman-Rachford ADI method for initial-boundary value problems, Mathematics of computation, 53, 187, 81-101, (1989) · Zbl 0689.65064
[18] Mattsson, K.; Nordström, J., Summation by parts operators for finite difference approximations of second derivatives, Journal of computational physics, 199, 503-540, (2004) · Zbl 1071.65025
[19] Carpenter, M.H.; Gottlieb, D.; Abarbanel, S., Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes, Journal of computational physics, 111, 220-236, (1994) · Zbl 0832.65098
[20] K. Mattsson, Private communication, 2005.
[21] Strand, B., Summation by parts for finite difference approximations for d/dx, Journal of computational physics, 110, 47-67, (1994) · Zbl 0792.65011
[22] Carpenter, M.H.; Nordström, J.; Gottlieb, D., A stable conservative interface treatment of arbitrary spatial accuracy, Journal of computational physics, 148, 341-365, (1999) · Zbl 0921.65059
[23] Kim, J.; Moin, P., Application of a fractional-step method to incompressible Navier-Stokes equations, Journal of computational physics, 59, 308-323, (1985) · Zbl 0582.76038
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.