A discontinuous \(hp\) finite element method for the Euler and Navier-Stokes equations. (English) Zbl 0985.76048

Baumann, Carlos Erik; Oden, J. Tinsley

Int. J. Numer. Methods Fluids 31, No. 1, 79-95 (1999).

Summary: This paper introduces a new method for the solution of Euler and Navier-Stokes equations, which is based on the application of a recently developed discontinuous Galerkin technique to obtain a compact, higher-order accurate and stable solver. The method involves a weak imposition of continuity conditions on the state variables and on inviscid and diffusive fluxes across inter-element and domain boundaries. Within each element the field variables are approximated using polynomial expansions with local support; therefore, this method is particularly amenable to adaptive refinements and polynomial enrichment. Moreover, the order of spectral approximation on each element can be adaptively controlled according to the regularity of the solution. The particular formulation on which the method is based makes possible a consistent implementation of boundary conditions, and the approximate solutions are locally (elementwise) conservative. The results of numerical experiments for representative benchmarks suggest that the method is robust, capable of delivering high rates of convergence, and well suited to be implemented in parallel computers.

MSC:

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

Keywords:

discontinuous \(hp\) finite element method; Navier-Stokes equations; Euler equations; compact stable solver; higher-order solver; discontinuous Galerkin technique; polynomial expansions

References:

[1]	Harten, SIAM J. Numer. Anal. 24 pp 279– (1987) · Zbl 0627.65102
[2]	Van Leer, J. Comp. Phys. 23 pp 175– (1977)
[3]	Woodward, J. Comput. Phys. 54 pp 174– (1984) · Zbl 0573.76057
[4]	’Error estimates for cell-vertex solutions of the compressible Euler equations’, Technical Report 87-6, ICASE, January, 1987.
[5]	and , ’Accuracy of schemes for the Euler equations with non-uniform meshes’, Technical Report 85-59, ICASE, December, 1985.
[6]	Babuška, Comput. Math. Appl. 37 pp 103– (1999) · Zbl 0940.65076
[7]	’A hp-adaptive discontinuous finite element method for computational fluid dynamics’, Ph.D. Dissertation, The University of Texas at Austin, August, 1997.
[8]	Baumann, Comput. Methods Appl. Mech. Eng. (1998)
[9]	Oden, J. Comput. Phys. 146 pp 491– (1998) · Zbl 0926.65109
[10]	Delves, J. Inst. Math. Appl. 23 pp 223– (1979) · Zbl 0443.65087
[11]	Nitsche, Abh. Math. Sem. Univ. Hamburg 36 pp 9– (1971) · Zbl 0229.65079
[12]	Bassi, J. Comput. Phys. (1997)
[13]	, and , ’The discontinuous Galerkin method applied to CFD problems’, 2nd Europ. Conf. on Turbomachinery, Fluid Dynamics and Thermodynamics, ASME, 1995.
[14]	Lomtev, Int. J. Numer. Methods Fluids (1997)
[15]	and , ’Simulations of viscous supersonic flows on unstructured meshes’, AIAA 97-0754, 1997.
[16]	Lomtev, J. Comput. Phys. (1998)
[17]	and , ’Spectral/hp methods for viscous compressible flows on unstructured 2D meshes’, Technical Report, Center for Fluid Mechanics Turbulence and Computation, Brown University, Providence, RI 02912, 1996.
[18]	, and , ’A discontinuous Galerkin method for the Navier-Stokes equations on hybrid grids’, Center for Fluid Mechanics 97-14, Division of Applied Mathematics, Brown University, 1997.
[19]	Cockburn, SIAM J. Numer. Anal. (1997)
[20]	The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
[21]	and , Texas Finite Elements Series, Vol. IV. Mathematical Aspects, Prentice-Hall, Englewood Cliffs, NJ, 1983.
[22]	and , An Introduction to the Mathematical Theory of Finite Elements, Wiley, New York, 1976.
[23]	’An hp-Adaptive discontinuous Galerkin method for hyperbolic conservation laws’, Ph.D. Dissertation, The University of Texas at Austin, May, 1994.
[24]	Cockburn, Math. Comp. 54 pp 545– (1990)
[25]	and , ’The Runge-Kutta discontinuous Galerkin method for conservation laws. V. Multi-dimensional systems’, ICASE Report 97-43, 1997.
[26]	Cockburn, J. Comput. Phys. 84 pp 90– (1989) · Zbl 0677.65093
[27]	Cockburn, Math. Comput. 52 pp 411– (1989)
[28]	Johnson, Math. Comput. 46 pp 1– (1986)
[29]	Lesaint, Numer. Math. pp 244– (1973) · Zbl 0283.65061
[30]	and , ’On a finite element method for solving the neutron transport equation’, in (ed.), Mathematical Aspects of Finite Elements in Partial Differential Equations, Academic Press, New York, 1974, pp. 89-123.
[31]	Lesaint, Math. Comput. 33 pp 891– (1979)
[32]	Babuška, Math. Model. Numer. Anal. 21 pp 199– (1987) · Zbl 0623.65113
[33]	and , ’Results of pressure distribution tests on a 0.010-scale space shuttle orbiter (61-0) in the Nasa/arc 3.5-foot hypersonic wind tunnel (test oh38)’, Technical Report, NASA, 1975.
[34]	Ghia, J. Comput. Phys. 48 pp 387– (1982) · Zbl 0511.76031