×

Asymptotic analyses for atmospheric flows and the construction of asymptotically adaptive numerical methods. (English) Zbl 1050.76056

From the conclusions: The “traditional” aim of scale analysis in meteorology is to obtain simplified asymptotic limit equations that are easier to solve and understand than the more comprehensive fully compressible flow equations, in the context of atmosphere flow modelling. We propose to construct new classes of “asymptotically adaptive numerical methods”, which do solve the full three-dimensionial compressible flow equations, but use the results of asymptotic scale analysis in the design of the discretizations. Such a scheme would assess a small number of nondimensional characteristic numbers “on the fly” during a computation. These characteristic numbers are chosen so as to indicate whether the current flow state is or is not within the vicinity of a singular limit regime. As a singular limit is approached, the discretizations automatically adapt, and they merge into a scheme for asymptotic limit equations when the limit is actually achieved.

MSC:

76U05 General theory of rotating fluids
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
86A10 Meteorology and atmospheric physics
76M99 Basic methods in fluid mechanics

Keywords:

singular limit
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abarbanel, Computers Fluids 17 pp 1– (1989) · Zbl 0664.76088 · doi:10.1016/0045-7930(89)90003-0
[2] ; ; ; ; : A conservative adaptive projection method for the variable density incompressible navier-stokes equations. LBNL Preprint 39075 UC-405 (1996).
[3] Batchelor, Quart. J. Roy. Meteorolog. Soc. 79 pp 224– (1953) · doi:10.1002/qj.49707934004
[4] Bijl, J. Comput. Phys. 141 pp 153– (1998) · Zbl 0918.76054 · doi:10.1006/jcph.1998.5914
[5] ; ; : Dry atmosphere asymptotics. Techn. Rep., Potsdam Institute for Climate Impact Research, Potsdam, Germany, 1999.
[6] Casulli, Int. J. Num. Meth. Fluids 4 pp 11– (1984) · Zbl 0549.76050 · doi:10.1002/fld.1650041102
[7] Chorin, J. Comput. Phys. 2 pp 12– (1967) · Zbl 0149.44802 · doi:10.1016/0021-9991(67)90037-X
[8] Chorin, Math. Comp. 22 pp 745– (1968) · doi:10.1090/S0025-5718-1968-0242392-2
[9] Durran, J. Atmospheric Sci. 46 pp 1453– (1989) · doi:10.1175/1520-0469(1989)046<1453:ITAA>2.0.CO;2
[10] : Numerical methods for wave equations in geophysical fluid dynamics. Springer, 1999.
[11] Ebin, I. Comm. Pure Appl. Math. 35 pp 451– (1982) · Zbl 0478.76011 · doi:10.1002/cpa.3160350402
[12] : Atmospheric convection. Oxford University Press, 1994.
[13] Embid, Comm. Partial Diff. Equ. 21 pp 619– (1996) · Zbl 0849.35106 · doi:10.1080/03605309608821200
[14] Embid, Theor. Comput. Fluid Dynamics 11 pp 155– (1998) · Zbl 0923.76339 · doi:10.1007/s001620050086
[15] : Erweiterung eines Godunov-Typ-Verfahrens für mehrdimensionale kompressible Strömungen auf die Fälle kleiner und verschwindender Machzahl. PhD thesis, RWTH Aachen 1998.
[16] ; ; ; : Multiple pressure variable (MPV) approach for low Mach number flows based on asymptotic analysis. In: Hirschel, E. H. (ed.): Flow simulation with high-performance computers II. DFG priority research programme results. Notes on Numerical Fluid Mechanics 52. Vieweg Verlag, Braunschweig 1996.
[17] : Atmosphere-ocean dynamics. International Geophysical Series 30. Academic Press, London 1982.
[18] Guillard, Computers Fluids 28 pp 63– (1999) · Zbl 0963.76062 · doi:10.1016/S0045-7930(98)00017-6
[19] Harlow, Phys. Fluids 8 pp 2182– (1965) · Zbl 1180.76043 · doi:10.1063/1.1761178
[20] ; : Asymptotic analysis and the numerical solution of parial differential equations. Lecture Notes in Pure and Applied Mathematics 130. Marcel Dekker, New York 1991.
[21] Karki, AIAA J. 27 pp 1167– (1989) · doi:10.2514/3.10242
[22] Klainerman, Comm. Pure Appl. Math. 35 pp 629– (1982) · Zbl 0478.76091 · doi:10.1002/cpa.3160350503
[23] Klein, I: One-dimensional flow. J. Comput. Physics 121 pp 213– (1995) · Zbl 0842.76053
[24] ; ; ; ; ; ; ; : Asymptotic adaptive methods for multi-scale problems in fluid mechanics. J. Engin. Math. (to appear). · Zbl 1015.76071
[25] Lai, AIAA Paper 3369 (1993)
[26] LeVeque, SIAM J. Num. Anal. 22 pp 151– (1985) · Zbl 0584.65058 · doi:10.1137/0722063
[27] : Numerical methods for conservation laws. Birkhäuser Verlag, Basel 1992. · Zbl 0847.65053
[28] Majda, Combustion Sci. Technol. 42 pp 185– (1985) · doi:10.1080/00102208508960376
[29] Merkle, AIAA J. 25 pp 831– (1987) · doi:10.2514/3.9708
[30] Muraki, J. Atmospheric Sci. 56 pp 1547– (1998) · doi:10.1175/1520-0469(1999)056<1547:TNOCTQ>2.0.CO;2
[31] Ogura, J. Atmosphere Sci. 19 pp 173– (1962) · doi:10.1175/1520-0469(1962)019<0173:SAODAS>2.0.CO;2
[32] : Geophysical fluid dynamics. 2nd ed. Springer, 1987.
[33] ; ; ; : The extension of incompressible flow solvers to the weakly compressible regime. Techn. Rep., Inst. f. Aerodynamik und Gasdynamik, Univ. Stuttgart 1999.
[34] : Boundary layer theory. MacGraw-Hill, 1968.
[35] Schneider, J. Comput. Physics 155 pp 248– (1999) · Zbl 0968.76054 · doi:10.1006/jcph.1999.6327
[36] : Mathematische Methoden in der Strömungsmechanik. Vieweg, 1978. · Zbl 0369.76001
[37] Schochet, J. Diff. Equ. 114 pp 476– (1994) · Zbl 0838.35071 · doi:10.1006/jdeq.1994.1157
[38] Sesterhenn, Comput. Fluid Dynamics 2 pp 829– (1992)
[39] : Why can’t stably stratified air rise over high ground. In: Blumen, W. (ed.): Atmospheric processes over complex terrain. Meteorological Monographs 23. 1990, pp. 105–107.
[40] Turkel, J. Comput. Physics 72 pp 277– (1987) · Zbl 0633.76069 · doi:10.1016/0021-9991(87)90084-2
[41] : Perturbation methods in fluid mechanics. Annotated Ed. Parabolic Press, 1975.
[42] : On the use and accuracy of compressible flow codes at low mach numbers. AIAA Paper 91-1662, 1991.
[43] Worlikar, J. Comput. Physics 144 pp 299– (1998) · Zbl 0920.76061 · doi:10.1006/jcph.1997.5816
[44] : Meteorological fluid dynamics. Lecture Notes Phys. m5. Springer, 1991.
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.