{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:28:07Z","timestamp":1787333287388,"version":"build-2736575974"},"reference-count":29,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>This article presents a new approach to solve the transonic flow problem by the algebraic multigrid (AMG) method. The proposed algorithm is demonstrated to attain excellent convergence rates, independent of the problem size, through the entire range of flow regimes. The mathematical difficulties of the problem are associated with the fact that the governing equation changes its type from elliptic (subsonic flow) to hyperbolic (supersonic flow). A pointwise relaxation method when applied directly to the upwind discrete operator, in the supersonic flow regime, is unstable. Resolving this difficulty is the main achievement of this work. A variety of issues regarding the AMG coarsening and construction of transfer operators is addressed in order to achieve the required efficiency for the problems under consideration. We demonstrate the AMG performance on a variety of model problems involving the quasi-linear full potential equation in two dimensions for subsonic, sonic, and supersonic flow and various flow directions (with respect to the grid). In addition, we present a preliminary result verifying the capabilities of the constructed algorithm to deal with a nonlinear problem\u2014transonic small-disturbance equation.<\/jats:p>","DOI":"10.1137\/090757162","type":"journal-article","created":{"date-parts":[[2010,7,6]],"date-time":"2010-07-06T18:04:54Z","timestamp":1278439494000},"page":"2007-2028","source":"Crossref","is-referenced-by-count":1,"title":["Toward Applying Algebraic Multigrid to Transonic Flow Problem"],"prefix":"10.1137","volume":"32","author":[{"given":"Shlomy","family":"Shitrit","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"David","family":"Sidilkover","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,7,6]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1016\/0041-5553(66)90118-2"},{"key":"R2","unstructured":"A. Brandt, S. F. McCormick, and J. W. Ruge,\n                      Algebraic multigrid (AMG) for sparse matrix equations\n                      , in Sparsity and Its Applications, Cambridge University Press, Cambridge, 1984."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(92)90049-5"},{"key":"R4","doi-asserted-by":"crossref","unstructured":"A. Brandt,\n                      Multi-level adaptive technique (mlat) for fast numerical solutions to boundary value problems\n                      , in Proceedings of the International Conference on Numerical Methods in Fluids Mechanics, 1973, pp. 82\u201389.","DOI":"10.1007\/BFb0118663"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1977-0431719-X"},{"key":"R6","unstructured":"A. Brandt,\n                      Stages in developing multigrid solutions\n                      , in Proceedings of the 2nd International Congress on Numerical Methods for Engineers, 1980, pp. 23\u201343."},{"key":"R7","unstructured":"A. Brandt,\n                      Multigrid Techniques: $1984$ Guide with Applications to Fluid Dynamics\n                      , The Weizmann Institute of Science, Rehovot, Israel, 1984."},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1016\/0096-3003(86)90095-0"},{"key":"R9","first-page":"1","volume":"10","author":"Brandt A.","year":"2000","journal-title":"Electron. Trans. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/1097-4067","issn-type":"print"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"W. L. Briggs, V. E. Henson, and S. F. McCormick,\n                      A Multigrid Tutorial\n                      , 2nd ed., SIAM, Philadelphia, 2000.","DOI":"10.1137\/1.9780898719505"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827596304046"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1996.0094"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1016\/0041-5553(62)90031-9"},{"key":"R14","unstructured":"W. Hackbusch,\n                      A multigrid method applied to a boundary problem with variable coefficients in a rectangle\n                      , Technical report 17-77, Institut f\u00fcr Angewandte Mathematik, Universit\u00e4t K\u00f6ln, 1977."},{"key":"R15","doi-asserted-by":"crossref","unstructured":"W. Hackbusch,\n                      Multigrid Methods and Applications\n                      , Springer, Heidelberg, 1985.","DOI":"10.1007\/978-3-662-02427-0"},{"key":"R16","doi-asserted-by":"crossref","unstructured":"A. Jameson, W. Schmidt, and E. Turkel,\n                      Numerical solution of the Euler equations by finite volume methods using Runge-Kutta time stepping Schemes\n                      , in AIAA 14th Fluid and Plasma Dynamics, Vol. 81-1259, AIAA, Reston, VA, 1981.","DOI":"10.2514\/6.1981-1259"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160270302"},{"key":"R18","unstructured":"A. Jameson,\n                      Transonic airfoil calculations using the Euler equations\n                      , in Numerical Methods in Aeronautical Fluid Dynamics, P. L. Roe, ed., Academic Press, New York, 1982."},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1016\/0096-3003(83)90019-X"},{"key":"R20","first-page":"355","volume":"15","author":"Kaczmarz S.","year":"1937","journal-title":"Bull. Acad. Pol. Sci. Lett. A"},{"key":"R21","doi-asserted-by":"crossref","unstructured":"B. Koren,\n                      Multigrid and defect correction for the steady Navier-Stokes equations\n                      , in Proceedings of the Fourth GAMM-Seminar on Robust Multi-Grid Methods, Kiel, 1988.","DOI":"10.1007\/978-3-322-86200-6_15"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.2514\/3.6131"},{"key":"R23","doi-asserted-by":"crossref","unstructured":"J. W. Ruge and K. Stueben,\n                      Algebraic multigrid\n                      , in Multigrid Methods, Vol. 3, S. F. McCormick, ed., SIAM, Philadelphia, 1987, pp. 73\u2013130.","DOI":"10.1137\/1.9781611971057.ch4"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-9274(01)00123-4"},{"key":"R25","unstructured":"J. C. South and A. Brandt,\n                      Application of a multi-level grid method to transonic flow calculations\n                      , in Transonic Flow Problems in Turbo-machinery, J. T. C. Adamson and M. F. Platzer, eds., Hemisphere, WA, 1977, pp. 180\u2013207."},{"key":"R26","unstructured":"S. Spekreijse,\n                      Multigrid solution of the steady Euler equations\n                      , Ph.D. thesis, CWI, Amsterdam, 1987."},{"key":"R27","unstructured":"K. Stueben,\n                      An Introduction to Algebraic Multigrid\n                      , Technical report, German National Research Center for Information Technology (GMD), San Diego, 2001."},{"key":"R28","unstructured":"A. S. U. Trottenberg and C. W. Oosterlee,\n                      Multigrid\n                      , Elsevier Academic Press, New York, 2001."},{"key":"R29","unstructured":"P. Wesseling,\n                      An Introduction to Multigrid Methods\n                      , John Wiley & Sons, New York, 1992."}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/090757162","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:48:02Z","timestamp":1787330882000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/090757162"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":29,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/090757162"],"URL":"https:\/\/doi.org\/10.1137\/090757162","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}