The solution of the compressible Euler equations at low Mach numbers using a stabilized finite element algorithm. (English) Zbl 1044.76036

Summary: We present a streamline-upwind/Petrov-Galerkin (SUPG) algorithm for the solution of compressible Euler equations at low Mach numbers. The Euler equations are written in terms of entropy variables which result in Jacobian matrices which are symmetric. We note that, in the low Mach number limit, the SUPG method with the standard choices for the stabilization matrix fail to provide adequate stabilization. This results in a degradation of the solution accuracy. We propose a stabilization matrix which incorporates dimensional-scaling arguments and which exhibits the appropriate behavior for low Mach numbers. To guide in the derivation of the new stabilization matrix, the non-dimensionalized equations are transformed to a new set of variables that converge, when the characteristic. Mach number tends to zero, to the incompressible velocity and pressure variables. The resulting algorithm is capable of accurately computing external flows with free stream Mach numbers as low as \(10^{-3}\).


76M10 Finite element methods applied to problems in fluid mechanics
76N15 Gas dynamics (general theory)


Full Text: DOI


[1] T.J. Barth, Numerical Methods for Gasdynamic Systems on Unstructured Meshes, Lecture Notes in Computational Science and Engineering, 1998, pp. 195-284 · Zbl 0969.76040
[2] Choi, D.; Merkle, C.L., Application of time-iterative schemes to incompressible flow, Aiaa j., 23, 1518-1524, (1985) · Zbl 0571.76016
[3] Choi, Y.H.; Merkle, C.L., The application of preconditioning in viscous flows, J. comput. phys., 105, 203-223, (1993) · Zbl 0768.76032
[4] Darmofal, D.L.; Schmid, P.J., The importance of eigenvectors for local preconditioners of the Euler equations, J. comput. phys., 127, 346-362, (1996) · Zbl 0860.76054
[5] Darmofal, D.L.; Van Leer, B., Local preconditioning: manipulating mother nature to fool father time, (), 211-239 · Zbl 0988.76059
[6] D.R. Fokkema, Subspace methods for linear, nonlinear and eigenproblems, Ph.D. Thesis, Utrech University, 1996
[7] Guillard, H.; Viozat, C., On the behavior of upwind schemes in the low Mach number limit, Comput. fluids, 28, 63-86, (1999) · Zbl 0963.76062
[8] Guerra, J.; Gustafsson, B., A numerical method for incompressible and compressible flow problems with smooth solutions, J. comput. phys., 63, 377-397, (1986) · Zbl 0608.76055
[9] Gustaffson, B., Unsymmetric hyperbolic systems and the Euler equations at low Mach numbers, J. sci. comput., 2, 2, (1987)
[10] Gustafsson, B.; Stoor, H., Navier – stokes equations for almost incompressible flow, SIAM J. numer. anal., 28, 1523-1547, (1991) · Zbl 0734.76048
[11] Harten, A., On the symmetric for of systems of conservation laws with entropy, J. comput. phys., 49, 151-164, (1983) · Zbl 0503.76088
[12] Hughes, T.J.R.; Franca, L.P.; Mallet, M., A new finite element formulation for computational fluid dynamics: I. symmetric forms of the compressible Euler and navier – stokes equations and the second law of thermodynamics, Comp. methods appl. mech. engrg., 54, 223-234, (1986) · Zbl 0572.76068
[13] Hughes, T.J.R.; Mallet, M., A new finite element formulation for computational fluid dynamics: III. the generalized streamline operator for multidimensional advective diffusive systems, Comput. methods appl. mech. engrg., 58, 305-328, (1986) · Zbl 0622.76075
[14] Hughes, T.J.R.; Franca, L.P.; Hulbert, G.M., A new finite element formulation for computational fluid dynamics: VIII. the Galerkin least-squares method for advective diffusive equations, Comput. methods appl. mech. engrg., 73, 173-189, (1989) · Zbl 0697.76100
[15] Klainerman, S.; Majda, A., Compressible and incompressible fluids, Commun. pure appl. math., 34, 629-651, (1981) · Zbl 0478.76091
[16] L. Machiels, A.T. Patera, J. Peraire, Y. Maday, A general framework for finite element of posteriori error control: Application to linear anad nonlinear convection-dominated problems, in: ICFD Conference on Numerical Methods for Fluid Dynamics, Oxford, England, March 31-April 3, 1998
[17] C.L. Reed, D.A. Anderson, Application of low speed preconditioning to the compressible Navier-Stokes equations, AIAA Paper 97-0873, 1997
[18] Roe, P.L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. comput. phys., 43, 357-372, (1981) · Zbl 0474.65066
[19] Shakib, F.; Hughes, T.J.R.; Johan, Z., A new finite element formulation for computational fluid dynamics: X. the compressible Euler and navier – stokes equations, Comput. methods appl. mech. engrg., 89, 141-219, (1991) · Zbl 0838.76040
[20] Sleijpen, G.L.G.; van der Vorst, H.A.; Fokkema, D.R., Bicgstab(1) and other hybrid bi-CG methods, Numer. algorithms, 7, 75-109, (1994) · Zbl 0810.65027
[21] Turkel, E., Symmetrization of fluid dynamic matrices with application, Math. comput., 27, 729-736, (1973) · Zbl 0298.65064
[22] Turkel, E.; Fiterman, A.; Van Leer, B., Preconditioning and the limit to the incompressible flow equations for finite differences schemes, (), 215-234
[23] B. Van Leer, W.T. Lee, P.L. Roe, Characteristic time-stepping or local preconditioning of the Euler equations, AIAA Paper 91-1552, 1991
[24] Weiss, J.M.; Smith, W.A., Preconditioning applied to variable and constant density flows, Aiaa j., 33, 2050-2057, (1995) · Zbl 0849.76072
[25] J.S. Wong, D.L. Darmofal, J. Peraire, High-order finite element discretization for the compressible Euler and Navier-Stokes equations, FDRL TR-01-1, Dept. of Aeronautics & Astronautics, MIT, April, 2001
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.