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Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows. II: Efficient flux quadrature. (English) Zbl 1099.76521
Summary: A new and efficient quadrature rule for the flux integrals arising in the space-time discontinuous Galerkin discretization of the Euler equations in a moving and deforming space-time domain is presented and analyzed. The quadrature rule is a factor three more efficient than the commonly applied quadrature rule and does not affect the local truncation error and stability of the numerical scheme. The local truncation error of the resulting numerical discretization is determined and is shown to be the same as when product Gauss quadrature rules are used. Details of the approximation of the dissipation in the numerical flux are presented, which render the scheme consistent and stable. The method is successfully applied to the simulation of a three-dimensional, transonic flow over a deforming wing.
Part I, cf. J. Comput. Phys. 182, No. 2, 546–585 (2002; Zbl 1057.76553).

76M10 Finite element methods applied to problems in fluid mechanics
76N15 Gas dynamics (general theory)
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[1] Adams, R.A., Sobolev spaces, (1975), Academic Press London · Zbl 0186.19101
[2] Atkins, H.L.; Shu, C.W., Quadrature-free implementation of discontinuous Galerkin method for hyperbolic equations, Am. inst. aeronaut. astronaut. J., 36, 5, 775-782, (1997)
[3] Bassi, F.; Rebay, S., High-order accurate discontinuous finite element solution of the 2D Euler equations, J. comput. phys., 138, 251-285, (1997) · Zbl 0902.76056
[4] Batten, P.; Clarke, N.; Lambert, C.; Causon, D.M., On the choice of wave speeds for the HLLC Riemann solver, SIAM J. sci. stat. comp., 18, 6, (1997) · Zbl 0992.65088
[5] Batten, P.; Leschziner, M.A.; Goldberg, U.C., Average-state Jacobians and implicit methods for compressible viscous and turbulent flows, J. comput. phys., 137, 38-78, (1997) · Zbl 0901.76043
[6] O.J. Boelens, H. van der Ven, B. Oskam, A.A. Hassan. The boundary conforming discontinuous Galerkin finite element approach for rotorcraft simulations, J. Aircraft, submitted for publication
[7] Bramble, J.H.; Hilbert, S.R., Bounds for a class of linear functional with applications to Hermite interpolation, Numer. math., 16, 362-369, (1971) · Zbl 0214.41405
[8] Brenner, S.C.; Scott, L.R., The mathematical theory of finite element methods, (1996), Springer Berlin
[9] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North-Holland Amsterdam · Zbl 0445.73043
[10] Cockburn, B.; Hou, S.; Shu, C.-W., The runge – kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Math. comp., 54, 545-581, (1990) · Zbl 0695.65066
[11] Grisvard, P., Elliptic problems in nonsmooth domains, (1985), Pitman London · Zbl 0695.35060
[12] Lions, J.-L., Sur LES espaces d’interpolation; dualité, Math. scand., 9, 147-177, (1961) · Zbl 0103.08102
[13] Lockard, D.P.; Atkins, H.L., Efficient implementations of the quadrature-free discontinuous Galerkin method, Am. inst. aeronaut. astronaut. J., 99, 3309, (1999)
[14] Masud, A.; Hughes, T.J.R., A space – time Galerkin/least squares finite element formulation of the navier – stokes equations for moving domain problems, Comput. meth. appl. mech. engrg., 146, 91-126, (1997) · Zbl 0899.76259
[15] Nikol’skii, S.M., On imbedding, continuation and approximation theorems for differentiable functions of several variables, Russian mat. surveys, 16, 55, (1961) · Zbl 0117.29101
[16] Toro, E.F., Riemann solvers and numerical methods for fluid dynamics, (1997), Springer Berlin · Zbl 0888.76001
[17] Triebel, H., Interpolation theory, function spaces, differential operators, (1978), North-Holland Amsterdam · Zbl 0387.46032
[18] J.J.W. van der Vegt, Geometric conditions on the invertibility and interpolation error of isoparametric hexahedral space – time elements, Math. Comp., submitted for publication
[19] van der Vegt, J.J.W.; van der Ven, H., Discontinuous Galerkin finite element method with anisotropic local grid refinement for inviscid compressible flows, J. comput. phys., 141, 46-77, (1998) · Zbl 0939.76048
[20] J.J.W. van der Vegt, H. van der Ven, Slip boundary conditions in discontinuous Galerkin discretizations of the Euler equations of gas dynamics. In Proceedings Fifth World Congress on Computational Mechanics, Vienna, July 7-12, 2002. Available from <http://wccm.tuwien.ac.at/> · Zbl 1057.76553
[21] J.J.W. van der Vegt, H. van der Ven, Space – time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows. Part I. General formulation, J. Comput. Phys., in press · Zbl 1057.76553
[22] van der Ven, H.; van der Vegt, J.J.W., Accuracy, resolution, and computational complexity of a discontinuous Galerkin finite element method, (), 439-444 · Zbl 1041.76546
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