Development of the collocations and least squares method

V. I. Isaev ; V. P. Shapeev

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 14 (2008) no. 1, pp. 41-60

Résumé

We propose and implement new, more general versions of the method of collocations and least squares (the CLS method) and, for a system of linear algebraic equations, an orthogonal method for accelerating the convergence of an iterative solution. The use of the latter method and the proper choice of values of control parameters, based on the results of investigating the dependence of the properties of the CLS method on these parameters, as well as some other improvements of the CLS method suggested in this paper, allow one to solve numerically problems for Navier–Stokes equations in a reasonable time using a single-processor computer even for grids as large as $1280\times1280$. In this case, the total number of unknown variables is $\sim25\cdot106$. The numerical results for the problem of the lid-driven cavity flow of a viscous fluid are in good agreement with known results of other authors, including those obtained by means of schemes of higher approximation order with a small artificial viscosity. This and some other facts prove that the new versions of the CLS method make it possible to obtain an approximate solution with high accuracy.

Bibliographie

[1] Carey G. G., Cheung Y. K., Lau S. L., “Mixed operator problems using least squares finite element collocation”, Comput. Methods Appl. Mech. Eng., 22 (1980), 121–130 | DOI | MR | Zbl

[2] Mizusawa T., Kajita T., “Application of point least squares method with B-splines in solid mechanics problems”, Int. J. Numer. Methods Eng., 18:6 (1982), 897–907 | DOI | Zbl

[3] Burgess G., Mahajerin E., “Rotational fluid flow using a least squares collocation technique”, Comput. Fluids, 12 (1984), 311–317 | DOI | MR | Zbl

[4] Dong S. B., Lopez A. E., “Natural vibrations of a clamped circular plate with rectilinear orthotropy by least-squares collocation”, Int. J. Solids Struct., 21 (1985), 515–526 | DOI | Zbl

[5] Jiang B.-N., Carey G. F., “Adaptive refinement for least-squares finite elements with element-by-element conjugate gradient solution”, Int. J. Numer. Meth. Engineering, 24 (1987), 569–580 | DOI | MR | Zbl

[6] Bentley L. R., Pinder G. F., Herrera I., “Solution of the advective-dispersive transport equation using a least squares collocation, Eulerian–Lagrangian method”, Numer. Methods Partial Differ. Equations, 5:3 (1989), 227–240 | DOI | MR | Zbl

[7] Semin L. G., Sleptsov A. G., Shapeev V. P., “Metod kollokatsii – naimenshikh kvadratov dlya uravnenii Stoksa”, Vychisl. tekhnologii, 1:2 (1996), 90–98 | MR | Zbl

[8] Karamyshev V., Kovenya V., Sleptsov A., “Adaptive methods for Navier–Stokes equations”, Proc. of the Third ECCOMAS conference on computational fluid dynamics, Vol. 1, John Wiley Sons, Paris, 1996, 301–307

[9] Semin L. G., Shapeev V. P., “Kollokatsionno-setochnyi metod resheniya kraevykh zadach dlya uravnenii Nave–Stoksa”, Matematicheskie problemy mekhaniki sploshnykh sred, Tez. dokl. sib. shkoly-seminara, Novosibirsk, 1997, 125–126

[10] Semin L. G., Shapeev V. P., “Metod kollokatsii i naimenshikh kvadratov dlya uravnenii Nave–Stoksa”, Vychisl. tekhnologii, 3:3 (1998), 72–84 | MR | Zbl

[11] Bochev P., Cai Z., Manteuffel T. A., McCormick S. F., “Analysis of velocity-flux first-order system least-squares principles for the Navier–Stokes equations. Part I”, SIAM J. Numer. Anal., 35:3 (1998), 990–1009 | DOI | MR | Zbl

[12] Semin L. G., Shapeev V. P., “Collocation-grid method for solving boundary problems for Navier–Stokes equations”, Proc. of the international conference on the methods of aerophysical research., Part 2, Novosibirsk, 1998, 186–191

[13] Belyaev V. V., Shapeev V. P., “Metod kollokatsii i naimenshikh kvadratov na adaptivnykh setkakh v oblasti s krivolineinoi granitsei”, Vychisl. tekhnologii, 5:4 (2000), 12–21 | MR

[14] Shapeev V. P., Semin L. G., Belyaev V. V., “The collocation and least squares method for numerical solution of Navier–Stokes equations”, Proc. of the Steklov Inst. of Math., Suppl. 2, 2003, S115–S137 | MR

[15] Semin L. G., “Collocation and least squares method for 2D heat conductivity equation”, Vychisl. tekhnologii, 11:1 (2006), 18–25

[16] Isaev V. I., Shapeev V. P., Eremin S. A., “Issledovanie svoistv metoda kollokatsii i naimenshikh kvadratov resheniya kraevykh zadach dlya uravneniya Puassona i uravnenii Nave–Stoksa”, Vychisl. tekhnologii, 12:3 (2007), 53–70

[17] Issaev V. I., Shapeev V. P., “Investigation of the collocation and least squares method for solving boundary value problems for Navier–Stokes equations”, Proc. of the international conference on the methods of aerophysical research (ICMAR2007), Part 3, Novosibirsk, 2007, 147–152

[18] Russell R. D., Shampine L. F., “A collocation method for boundary value problems”, Numer. Math., 19 (1972), 1–28 | DOI | MR | Zbl

[19] Boor C. de, Swartz B., “Collocation at Gaussian points”, SIAM J. Numer. Anal., 10:4 (1973), 582–606 | DOI | MR | Zbl

[20] Ascher U., Christiansen J., Russel R. D., “A collocation solver for mixed order systems of boundary value problems”, Math. Comp., 33 (1979), 659–679 | DOI | MR | Zbl

[21] Sleptsov A. G., Letova E. Yu., Salomatov K. G., Shmykov I. V., “Pereopredelennye sistemy konechno-raznostnykh uravnenii dlya zadach tipa diffuzii-konvektsii”, Vychisl. tekhnologii, 2:5 (1993), 192–201 | MR

[22] Shapeev A. V., “O dvukh algoritmakh chislennykh metodov lineinoi algebry”, Student i nauchno-tekhnicheskii progress, Tez. dokl. mezhdunar. nauch. stud. konf., Novosibirsk, 2000, 16

[23] Moffat H. K., “Viscous and resistive eddies near a sharp corner”, J. Fluid Mech., 18 (1964), 1–18 | DOI | Zbl

[24] Barragy E., Carey G. F., “Stream function-vorticity driven cavity solution using $p$ finite elements”, Computers Fluids, 26:5 (1997), 453–468 | DOI | Zbl

[25] Garanzha V. A., Konshin V. N., “Chislennye algoritmy dlya techenii vyazkoi zhidkosti, osnovannye na konservativnykh kompaktnykh skhemakh vysokogo poryadka approksimatsii”, Zhurn. vychisl. matematiki i mat. fiziki, 39:8 (1999), 1378–1392 | MR | Zbl

[26] Beklemishev D. V., Dopolnitelnye glavy lineinoi algebry, Nauka, M., 1983 | MR | Zbl

[27] Temam R., Uravneniya Nave–Stoksa. Teoriya i chislennyi analiz, Mir, M., 1981 | MR | Zbl

[28] Ladyzhenskaya O. A., Matematicheskie voprosy dinamiki vyazkoi neszhimaemoi zhidkosti, Nauka, M., 1970 | MR

[29] Saad Y., “Krylov subspace methods for solving large unsymmetric linear systems”, Mathematics of Computation, 37:155 (1981), 105–126 | DOI | MR | Zbl

[30] Sleptsov A. G., “Ob uskorenii skhodimosti lineinykh iteratsii”, Modelirovanie v mekhanike, 3(20), no. 3, Novosibirsk, 1989, 132–147 | MR | Zbl

[31] Sleptsov A. G., “Ob uskorenii skhodimosti lineinykh iteratsii 2”, Modelirovanie v mekhanike, 3(20), no. 5, Novosibirsk, 1989, 132–147 | MR | Zbl

[32] Fedorenko R. P., Vvedenie v vychislitelnuyu fiziku, MFTI, M., 1994

[33] Karamyshev V. B., Kovenya V. M., Sleptsov A. G., Cherny S. G., “Variational method of accelerating linear iterations and its applications”, Computers Fluids, 25:5 (1996), 467–484 | DOI | MR | Zbl

[34] Olshanskii M. A., Ravnomernye po parametru mnogosetochnye i iteratsionnye metody, Dis. $\dots$ dokt. fiz.-mat. nauk, M., 2006

[35] Shapeev A. V., “Raznostnaya skhema chetvertogo poryadka dlya uravnenii Nave–Stoksa na pyatitochechnom shablone”, Dinamika sploshnoi sredy, 116, Novosibirsk, 2000, 119–122 | MR | Zbl

[36] Ghia U., Ghia K. N., Shin C. T., “High-Re solutions for incompressible flow using the Navier–Stokes equations and a multigrid method”, J. Comput. Phys., 48 (1982), 387–411 | DOI | Zbl

[37] Chen C. J., Chen H. J., “Finite analytic numerical method for unsteady two-dimensional Navier–Stokes equations”, J. Comput. Physics, 53:2 (1984), 209–226 | DOI | Zbl

[38] Bruneau C. H., Jouron C., “An efficient scheme for solving steady incompressible Navier–Stokes equations”, J. Comput. Physics, 89:2 (1990), 389–413 | DOI | MR | Zbl

[39] Deng G. B., Piquet J., Queutey P., Visonneau M., “A new fully coupled solution of the Navier–Stokes equations”, Int. J. for Numer. Methods in Fluids, 19:7 (1994), 605–639 | DOI | MR | Zbl

[40] Botella O., Peyret R., “Benchmark spectral results on the lid-driven cavity flow”, Computers Fluids, 27:4 (1998), 421–433 | DOI | MR | Zbl

[41] Shankar P. N., Deshpande M. D., “Fluid mechanics in the driven cavity”, Annu. Rev. Fluid Mech., 32 (2000), 93–136 | DOI | MR | Zbl

[42] Sahin M., Owens R. G., “A novel fully implicit finite volume method applied to the lid-driven cavity problem. Part I: High Reynolds number flow calculations”, Int. J. Numer. Meth. Fluids, 42 (2003), 57–77 | DOI | MR | Zbl

[43] Wu Y., Liao S., “Solving high Reynolds-number viscous flows by the general BEM and domain decomposition method”, Int. J. Numer. Meth. Fluids, 47 (2005), 185–199 | DOI | Zbl

[44] Erturk E., Corke T. C., Gokcol C., “Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers”, Int. J. Numer. Meth. Fluids, 48:7 (2005), 747–774 | DOI | Zbl

[45] Cheng M., Hung K. C., “Vortex structure of steady flow in a rectangular cavity”, Computers Fluids, 35 (2006), 1046–1062 | DOI

[46] Chetverushkin B. N., Introductory word and plenary talk, V International Congress on Mathematical Modelling. Dubna, 2002

[47] Tolstykh A. I., Kompaktnye raznostnye skhemy i ikh primenenie v zadachakh aerogidrodinamiki, Nauka, M., 1990 | MR