Geometric and algebraic multigrid techniques for fluid dynamics problems on unstructured grids

K. N. Volkov   ; V. N. Emel'yanov   ; I. V. Teterina

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 2, pp. 283-300

Résumé

Issues concerning the implementation and practical application of geometric and algebraic multigrid techniques for solving systems of difference equations generated by the finite volume discretization of the Euler and Navier–Stokes equations on unstructured grids are studied. The construction of prolongation and interpolation operators, as well as grid levels of various resolutions, is discussed. The results of the application of geometric and algebraic multigrid techniques for the simulation of inviscid and viscous compressible fluid flows over an airfoil are compared. Numerical results show that geometric methods ensure faster convergence and weakly depend on the method parameters, while the efficiency of algebraic methods considerably depends on the input parameters.

Bibliographie

[1] Brandt A., “Guide to multigrid development”, Lect. Notes in Math., 960, 1982, 220–312 | DOI | MR | Zbl

[2] Wesseling P., An introduction to mutigrid methods, Wiley, Chichester, 1992 | MR

[3] Pierce N. A., Giles M. B., Jameson A., Martinelli L., Accelerating three-dimensional Navier–Stokes calculations, AIAA Paper No 97-1953, 1997

[4] Volkov K. N., “Blochnoe predobuslovlivanie uravnenii Eilera i Nave–Stoksa pri modelirovanii nizkoskorostnykh techenii na nestrukturirovannykh setkakh”, Zh. vychisl. matem. i matem. fiz., 49:10 (2009), 1868–1884 | MR | Zbl

[5] Volkov K. N., “Mnogosetochnye tekhnologii dlya resheniya zadach gazovoi dinamiki na nestrukturirovannykh setkakh”, Zh. vychisl. matem. i matem. fiz., 50:11 (2010), 1938–1952 | MR | Zbl

[6] Mavriplis D. J., “An assessment of linear versus nonlinear multigrid methods for unstructured mesh solvers”, J. Comput. Phys., 175:1 (2002), 302–325 | DOI | Zbl

[7] Brandt A., “Multi-level adaptive solutions to boundary value problems”, Math. Comput., 31:138 (1977), 333–390 | DOI | MR | Zbl

[8] Trottenberg U., Oosterlee C., Schüller A., Multigrid, Acad. Press, London, 2001 | MR | Zbl

[9] Wesseling P., Oosterlee C. W., “Geometric multigrid with applications to computational of fluid dynamics”, J. Comput. Appl. Math., 128:1–2 (2001), 311–334 | DOI | MR | Zbl

[10] Fedorenko R. P., “Relaksatsionnyi metod resheniya raznostnykh ellipticheskikh uravnenii”, Zh. vychisl. matem. i matem. fiz., 1:5 (1961), 922–927 | MR | Zbl

[11] Brannick J., Brezina M., Falgout R., Manteuffel T., McCormick S., Ruge J., Sheehan B., Xu J., Zikatanov L., “Extending the applicability of multigrid methods”, J. Phys: Conference Series, 46 (2006), 443–452 | DOI | MR

[12] Hackbusch W., Multigrid method and application, Springer-Verlag, Berlin, 1985 | MR

[13] Jameson A., “Solution of the Euler equations by a multigrid method”, Appl. Math. Comput., 13:3–4 (1983), 327–356 | DOI | MR

[14] Jameson A., “Analysis and design of numerical schemes for gas dynamics: artificial diffusion, upwind biasing, limiters and their effect on accuracy and multigrid convergence”, Internat. J. Comput. Fluid Dynamics, 4:3–4 (1995), 171–218 | DOI

[15] Crumpton P. I., Moinier P., Giles M. B., “An unstructured algorithm for high Reynolds number flows on highly stretched grids”, Numer. Meth. Laminar and Turbulent Flows, eds. C. Taylor, J. T. Cross, Pineridge Press, 1997, 561–572

[16] Müller J.-D., Giles M. B., “Edge-based multigrid schemes for hybrid grids”, Numerical Meth. Fluid Dynamics, 6 (1998), 425–432

[17] Cleary A. J., Falgout R. D., Henson V. E., Jones J. E., Manteu T. A., McCormick S. F., Miranda G. N., Ruge J. W., “Robustness and scalability of algebraic multigrid”, SIAM J. Sci. Statistical Comput., 21:6 (2000), 1886–1908 | DOI | MR | Zbl

[18] Ruge J., Stüben K., “Algebraic multigrid (AMG)”, Multigrid Methods, Frontiers in Appl. Math., 3, ed. S. F. McCormick, SIAM, Philadelphia, 1987, 73–130 | MR

[19] Stüben K., “A review of algebraic multigrid”, J. Computat. Appl. Math., 128:1–2 (2001), 281–309 | DOI | MR | Zbl

[20] Stüben K., “An introduction to algebraic multigrid”: U. Trottenberg, C. Oosterlee, A. Schüller, Multigrid, Academic Press, London, 2001, 413–532 | MR

[21] Yang U. M., “Parallel algebraic multigrid methods — high performance preconditioners”, Numerical Solution of Partial Differential Equations on Parallel Computers, Lecture Notes in Computational Science and Engineering, 51, eds. A. M. Bruaset, A. Tveito, Springer-Verlag, Berlin, 2006, 209–236 | DOI | MR | Zbl

[22] Volkov K. N., “Diskretizatsiya uravnenii Nave-Stoksa na nestrukturirovannoi setke pri pomoschi metoda kontrolnogo ob'ema i raznostnykh skhem povyshennoi razreshayuschei sposobnosti”, Zh. vychisl. matem. i matem. fiz., 48 (2008), 1250–1273 | MR

[23] AGARD Fluid Dynamics Panel. Test cases for inviscid flow field methods, AGARD Advisory Reports No AR-211, 1986