zbMATH — the first resource for mathematics

High order ADI method for solving unsteady convection-diffusion problems. (English) Zbl 1053.65067
Summary: We propose a high order alternating direction implicit (ADI) solution method for solving unsteady convection-diffusion problems. The method is fourth order in space and second order in time. It permits multiple use of the one-dimensional tridiagonal algorithm with a considerable saving in computing time, and produces a very efficient solver. It is shown through a discrete Fourier analysis that the method is unconditionally stable for 2D problems. Numerical experiments are conducted to test its high accuracy and to compare it with the standard second-order Peaceman-Rachford ADI method and the spatial third-order compact scheme of B. J. Noye and H. H. Tan [Int. J. Numer. Methods Eng. 26, No. 7, 1615–1629 (1988; Zbl 0638.76104)].

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Full Text: DOI
[1] D’yakonov, E.G., Difference schemes of second-order accuracy with a splitting operator for parabolic equations without mixed derivatives, Zh. vychisl. mat. I mat. fiz., 4, 935-941, (1964), (in Russian)
[2] Gupta, M.M.; Manohar, R.P.; Stephenson, J.W., A single cell high-order scheme for the convection – diffusion equation with variable coefficients, Int. J. numer. methods fluids, 4, 641-651, (1984) · Zbl 0545.76096
[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, Int. J. numer. methods fluids, 38, 1111-1131, (2002) · Zbl 1094.76546
[4] Karaa, S.; Zhang, J., Analysis of stationary iterative methods for discrete convection – diffusion equation with a nine-point compact scheme, J. comput. appl. math., 154, 447-476, (2003) · Zbl 1029.65119
[5] Noye, B.J.; Tan, H.H., A third-order semi-implicit finite difference method for solving the one-dimensional convection – diffusion equation, Int. J. numer. methods eng., 26, 1615-1629, (1988) · Zbl 0638.76104
[6] Noye, B.J.; Tan, H.H., Finite difference methods for solving the two-dimensional advection – diffusion equation, Int. J. numer. methods fluids, 26, 1615-1629, (1988) · Zbl 0638.76104
[7] Peaceman, D.W.; Rachford, H.H., The numerical solution of parabolic and elliptic differential equations, J. soc. ind. appl. math., 3, 28-41, (1959) · Zbl 0067.35801
[8] Rigal, A., Schémas compacts d’ordre élevé: application aux problémes bidimensionnels de diffusion – convection instationnaire I, C.R. acad. sci. Paris sr. I math., 328, 535-538, (1999) · Zbl 0936.65099
[9] Rigal, A., High order difference schemes for unsteady one-dimensional diffusion – convection problems, J. comput. phys., 114, 59-76, (1994) · Zbl 0807.65096
[10] P.J. Roach, Computational Fluid Dynamics, Hermosa, Albuquerque, NM, 1976
[11] Spotz, W.F.; Carey, G.F., Extension of high-order compact schemes to time-dependent problems, Numer. methods partial differential equations, 17, 657-672, (2001) · Zbl 0998.65101
[12] W.F. Spotz, High-order compact finite difference schemes for computational mechanics, Ph.D. Thesis, University of Texas at Austin, Austin, TX, 1995
[13] J. Zhang, Multigrid acceleration techniques and applications to the numerical solution of partial differential equations, Ph.D. Thesis, The George Washington University, Washington, DC, 1997
[14] Zhang, J., Multigrid method and fourth order compact difference scheme for 2D Poisson equation with unequal meshsize discretization, J. comput. phys., 179, 170-179, (2002) · Zbl 1005.65137
[15] Zhang, J.; Sun, H.; Zhao, J.J., High order compact scheme with multigrid local mesh refinement procedure for convection diffusion problems, Comput. methods appl. mech. eng., 191, 4661-4674, (2002) · Zbl 1068.76066
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.