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The characteristic-based-split procedure: An efficient and accurate algorithm for fluid problems. (English) Zbl 0985.76069

From the summary: We describe an algorithm designed to replace the Taylor-Galerkin (or Lax-Wendroff) methods, used by them so far in the solution of compressible flow problems. The new algorithm is applicable to a wide variety of situations, including fully incompressible flows and shallow water equations, as well as supersonic and hypersonic situations, and has proved to be always at least as accurate as other algorithms currently used. The algorithm is based on the solution of conservation equations of fluid mechanics to avoid any possibility of spurious solutions that may otherwise result. The main aspect of the procedure is to split the equations into two parts, (1) a part that is a set of simple scalar equations of convective-diffusion type for which it is well known that the characteristic Galerkin procedure yields an optimal solution; and (2) the part where the equations are self-adjoint and therefore discretized optimally by the Galerkin procedure. It is possible to solve both the first and second parts of the system explicitly, retaining there the time step limitations of the Taylor-Galerkin procedure. But it is also possible to use semi-implicit processes where in the first part we use a much bigger time step generally governed by the Peclet number of the system, while the second part is solved implicitly and is unconditionally stable. It turns out that the characteristic-based-split process allows equal interpolation to be used for all system variables without difficulties when the incompressible or nearly incompressible state is reached.

MSC:

76M25 Other numerical methods (fluid mechanics) (MSC2010)
76N15 Gas dynamics (general theory)
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[1] and , ’Search for a general fluid mechanics algorithm’, in and (eds.), Frontiers of Computational Fluid Dynamics, Wiley, New York, 1995, pp. 101-113.
[2] Zienkiewicz, Int. J. Numer. Methods Fluids 20 pp 869– (1995)
[3] Zienkiewicz, Int. J. Numer. Methods Fluids 20 pp 887– (1995)
[4] Codina, Int. J. Numer. Methods Fluids 27 pp 13– (1998)
[5] Satya Sai, Int. J. Numer. Methods Fluids 27 pp 57– (1998)
[6] Zienkiewicz, Int. J. Numer. Methods Fluids 23 pp 1– (1996)
[7] and , A fractional step method for the solution of compressible Navier-Stokes equations, in and (eds.), Computational Fluid Dynamics Review 1998, Vol. 1, World Scientific, Singapore, 1998, pp. 331-347.
[8] Zienkiewicz, Comput. Method Appl. Mech. Eng. 78 pp 105– (1990)
[9] ’Explicit or semi-explicit general algorithm for compressible and incompressible flows with equal finite element interpolation’, Report 90/5, Chalmers University of Technology, 1990.
[10] ’Finite elements and computational fluid mechanics’56-61, Metodos Numericos en Ingenieria, SEMNI Congress, Gran Canaria, 1991.
[11] Zienkiewicz, Int. J. Numer. Methods Eng. 32 pp 1184– (1991)
[12] Zienkiewicz, Int. J. Numer. Methods Eng. 35 pp 457– (1992)
[13] Lax, Comm. Pure. Appl. Math. 13 pp 217– (1960)
[14] Donea, Comput. Methods Appl. Mech. Eng. 45 pp 123– (1984)
[15] Donea, Int. J. Numer. Methods Eng. 20 pp 101– (1984)
[16] Löhner, Int. J. Numer. Methods Fluids 4 pp 1043– (1984)
[17] , and , High speed compressible flow and other advection dominated problems of fluid mechanics, in , and (eds.), Finite Elements in Fluids, vol. 6, Chap. 2Wiley, New York, 1985, pp. 41-88.
[18] Chorin, Math. Comput. 22 pp 745– (1968)
[19] Chorin, Math. Comput. 23 pp 341– (1969)
[20] Goudreau, Comput. Methods Appl. Mech. Eng. 33 pp 725– (1982)
[21] Zienkiewicz, Int. J. Numer. Methods Eng. 43 pp 565– (1998)
[22] Peraire, Int. J. Numer. Methods Eng. 22 pp 547– (1986)
[23] Zienkiewicz, Int. J. Numer. Methods Fluids 20 pp 1061– (1995)
[24] and , ’An improved finite element model for shallow water problems’, in (ed.), Finite Element Modelling of Environmental Problems, Wiley, New York, 1996, pp. 61-84.
[25] and , ’Tide and bore propagation in the Severn Estuary by a new fluid algorithm’1543-1552, Proc. Int. Conf. on Finite Elements in Fluids–New Trends and Applications, Venezia, 1995.
[26] and , ’Finite element solution of effluent dispersion’, in and (eds.), Numerical Methods in Fluid Mechanics, Pentech Press, 1974, pp. 325-354.
[27] and , ’Multistep Galerkin methods along characteristics of convection-diffusion problems’, in and (eds.), Advanced Computer Methods in PDEs IV, IMACS, Rutgers University, 1981, pp. 28-36.
[28] Pironneau, Numer. Math. 38 pp 309– (1982)
[29] , and , Characteristics and finite element methods applied to equations of fluids, in (ed.), The Mathematics of Finite Elements and Applications, Vol. V, Academic Press, New York, 1982, pp. 471-478.
[30] Bercovier, Appl. Math. Model. 7 pp 89– (1983)
[31] , and , ’Application des methodes du decomposition aux calculs numeriques an hydraulique industrielle’, INRIA 4th Coll. Methodes de Calcul Sci. et Techn., Versailles, 1983.
[32] and , The Finite Element Method, Vol. 2, 4th edition, McGraw-Hill, New York, 1991.
[33] Oñate, Comput. Methods Appl. Mech. Eng. 151 pp 233– (1998)
[34] Codina, Int. J. Numer. Methods Eng. 36 pp 1445– (1993)
[35] and , ’Euler computations of AGARD working group 07 aerofoil test cases’, AIAA 23rd Aerospace Sciences Meeting, Reno, NV, 14-17 January, 1985.
[36] Nithiarasu, Int. J. Numer. Methods Fluids 28 pp 1325– (1998)
[37] ’Numerical solutions of the Navier-Stokes equations for the supersonic laminar flow over a two-dimensional compression corner’, NASA TR-R-385, 1972.
[38] Peraire, J. Comput. Phys. 72 pp 449– (1987)
[39] Zienkiewicz, Int. J. Numer. Methods Eng. 37 pp 2189– (1994)
[40] Borouchaki, Finite Elem. Anal. Des. 25 pp 61– (1997)
[41] and , ’Towards better choices of adaptive mesh generation for fluid mechanics problems’, 6th ACME-UK Conference, 6-7 April, 1998, pp. 129-132.
[42] Ghia, J. Comput. Phys. 48 pp 387– (1982)
[43] Denham, Trans. Inst. Chem. Eng. 52 pp 361– (1974)
[44] Papanastasiou, Int. J. Numer. Methods Fluids 14 pp 587– (1992)
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