## The analysis of unsteady incompressible flows by a three-step finite element method.(English)Zbl 0772.76036

This paper describes a three-step finite element method and its applications to unsteady incompressible fluid flows. Stability analysis of the one-dimensional pure convection equation shows that this method has third-order accuracy and an extended numerical stability domain in comparison with the Lax-Wendroff finite element method. The method is cost-effective for incompressible flows because it permits less frequent updates of the pressure field with good accuracy. In contrast with the Taylor-Galerkin method, the present method does not contain any new higher-order derivatives, which makes it suitable for solving nonlinear multidimensional problems and flows with complicated boundary conditions.

### MSC:

 76M10 Finite element methods applied to problems in fluid mechanics 76D05 Navier-Stokes equations for incompressible viscous fluids 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Full Text:

### References:

 [1] Brooks, Comput. Methods Appl. Mech. Eng. 32 pp 199– (1982) [2] and , ’A multi-dimensional upwind scheme with no crosswind diffusion’, in (ed.) Finite Element Methods for Convection Dominated Flows, AMD Vol. 34, ASME, New York, 1979. [3] Hughes, Comput. Methods Appl. Mech. Eng. 45 pp 217– (1984) [4] and , ’Development of time-accurate finite element techniques for first-order hyperbolic systems with particular emphasis on the compressible Euler equations’, Report Prepared under NASA-Ames University Consortium Interchange NCA2-0R745104, 1982. [5] and , ’Finite element formulations for convection dominated flows with particular emphasis on the compressible Euler equations’, AIAA Paper 83-0125, Proc. AIAA 21st Aerospace Sciences Meeting, Reno, NV, 1983. [6] Tezduyar, Comput. Methods Appl. Mech. Eng. 59 pp 49– (1985) [7] Donca, Int. j. numer. methods eng. 20 pp 101– (1984) [8] Donea, Comput. Methods Appl. Mech. Eng. 45 pp 123– (1984) [9] Donea, J. Comput. Phys. 70 pp 463– (1987) [10] Selmin, Comput. Methods Appl. Mech. Eng. 52 pp 817– (1985) [11] ’Convergence of finite element Lax-Wendroff method for linear hyperbolic differential equation’, Proc. JSCE, No 253, 95-107 (1976). [12] Lohner, Int. j. numer. methods fluids 4 pp 1043– (1984) [13] Laval, Int. j. numer. methods fluids 11 pp 501– (1990) [14] Jiang, Fluid Dyn. Res. 9 pp 165– (1992) [15] Gresho, Int. j. numer. methods fluids 4 pp 557– (1984) [16] Hawken, Int. j. numer. methods fluids 10 pp 327– (1990) [17] and , ’A finite element simulation of viscous flow around a cylinder’, Proc. Int. Conf. on Numerical Methods in Engineering: Theory and Applications, Swansea, January 1990, pp. 955-962. [18] and , ’Development of a transient approach to simulate Newtonian and non-Newtonian flow’, Proc. Int. Conf. on Numerical Methods in Engineering: Theory and Applications, Swansea, January 1990, pp. 1003-1012. [19] Gresho, Int. j. numer. methods fluids 7 pp 1111– (1987) [20] and , ’Interaction analysis between structure and fluid flow using the direct Laplacian method’, Proc. 4th Int. Conf. on Computing in Civil and Building Engineering, Tokyo, July 1991, pp. 267-274. [21] Nallasamy, J. Fluid Mech. 79 pp 391– (1977) [22] Sivaloganathan, Int. j. numer. methods fluids 8 pp 417– (1988) [23] Ghia, J. Comput. Phys. 48 pp 387– (1982) [24] Schreiber, J. Comput. Phys. 49 pp 310– (1983) [25] Daly, Phys. Fluids 11 pp 15– (1968) [26] Ramaswamy, J. Comput. Phys. 90 pp 396– (1990) [27] Kawahara, Int. j. numer. methods fluids 5 pp 981– (1987)
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.