Abstract
A general procedure to construct alternating direction implicit (ADI) schemes for multidimensional problems was originated by Beam and Warming, using the method of approximate factorization. The technique which can be combined with a high-order linear multistep (LM) method introduces a factorization error that is of order two in the time step Δt. Thus, the approximate factorization method imposes a second-order temporal accuracy limitation independent of the accuracy of the LM method chosen as the time differencing approximation. We introduce a correction term to the right-hand side of a factored scheme to increase the order of the factorization error in Δt, and recover the temporal order of the original scheme. The method leads in particular to the modified ADI scheme proposed by Douglas and Kim. A convergence proof is given for the improved scheme based on the BDF2 method.
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References
Beam M, Warming RF (1976) An implicit finite-difference algorithm for hyperbolic systems in conservation-law form. J Comput Phys 22: 87–110
Beam M, Warming RF (1980) Alternating direction implicit methods for parabolic equations with a mixed derivative. SIAM J Sci Stat Comput 1: 131–159
Briley WR, McDonald H (1980) On the structure and use of linearized block implicit schemes. J Comput Phys 34: 54–73
Briley WR, McDonald H (2001) An overview and generalization of implicit Navier-Stokes algorithms and approximate factorization. Comput Fluids 30: 807–828
Douglas J Jr (1955) On the numerical integration of \({\frac{\partial^2 u}{\partial x^2}+ \frac{\partial^2 u}{\partial y^2}=\frac{\partial u}{\partial t}}\) by implicit methods. J Soc Ind Appl Math 3: 42–65
Douglas J Jr, Gunn EJ (1964) A general formulation of alternating direction methods. I. Parabolic and hyperbolic problems. Numer Math 6: 428–453
Douglas J Jr, Kim S (2001) Improved accuracy for locally one-dimensional methods for parabolic equations. Math Models Methods Appl Sci 11: 1563–1579
Douglas J Jr, Rachford HH (1956) On the numerical solution of heat conduction problems in two and three space variables. Trans Am Math Soc 82: 421–439
Hundsdorfer W (1998) A note on the stability of Douglas splitting method. Math Comp 67: 183–190
Karaa S (2009) A hybrid padé ADI scheme of higher-order For convection-diffusion problems. Int J Numer Meth Fluids (submitted)
Lambert JD (2000) Numerical methods for ordinary differential systems. Wiley, New York
Marchuk GI (1990) Splitting and alternating direction methods. Handbook of numerical analysis vol. I. North-Holland, Amsterdam, pp 197–462
Peaceman DW, Rachford HH Jr (1955) The numerical solution of parabolic and elliptic differential equations. J Soc Ind Appl Math 3: 28–41
van der Houwen PJ, Sommeijer BJ (2001) Approximate factorization for time-dependent partial differential equations. J Comput Appl Math 128: 447–466
Warming RF, Beam M (1978) On the construction and application of implicit factored schemes for conservation laws. In: Computational fluid dynamics (Proc. SIAM-AMS Sympos. Appl. Math., New York, (1977)), pp 85–129. SIAM-AMS Proc., vol XI. Amer Math Soc, Providence, RI
Warming RF, Beam M (1979) An extension of A-stability to alternating direct implicit methods. BIT 19: 1111–1131
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Communicated by C.C. Douglas.
This research was supported by Sultan Qaboos University under Grant IG/SCI/DOMS/07/08.
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Karaa, S. Improved accuracy for the approximate factorization of parabolic equations. Computing 86, 23–36 (2009). https://doi.org/10.1007/s00607-009-0063-6
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DOI: https://doi.org/10.1007/s00607-009-0063-6