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A semi-implicit discontinuous Galerkin finite element method for the numerical solution of inviscid compressible flow. (English) Zbl 1116.76386
Summary: The paper is concerned with the numerical solution of an inviscid compressible flow with the aid of the discontinuous Galerkin finite element method. Since the explicit time discretization requires a high restriction of the time step, we propose semi-implicit numerical schemes based on the homogeneity of inviscid fluxes, allowing a simple linearization of the Euler equations which leads to a linear algebraic system on each time level. Numerical experiments performed for the Ringleb flow problem verify a higher order of accuracy of the presented method and demonstrate lower CPU-time costs in comparison with an explicit method. Then the method is tested on more complex unsteady Euler flows.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
Software:
HE-E1GODF
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[1] Adjerid, S.; Devine, D.; Flaherty, J.E.; Krivodonova, L., A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems, Comput. methods appl. mech. engrg., 191, 1097-1112, (2002) · Zbl 0998.65098
[2] 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
[3] Bassi, F.; Rebay, S., A high order discontinuous Galerkin method for compressible turbulent flow, (), 113-123 · Zbl 0991.76039
[4] Baumann, C.E.; Oden, J.T., A discontinuous hp finite element method for the Euler and navier – stokes equations, Int. J. numer. meth. fluids, 31, 79-95, (1999) · Zbl 0985.76048
[5] Beam, R.M.; Warming, R.F., An implicit finite-difference algorithm for hyperbolic systems in conservation-law form, J. comput. phys., 22, 87-110, (1976) · Zbl 0336.76021
[6] Beam, R.M.; Warming, R.F., An implicit factored scheme for the compressible navier – stokes equations, Aiaa j., 16, 393-402, (1978) · Zbl 0374.76025
[7] Bejček, M.; Dolejšı́, V.; Feistauer, M., On discontinuous Galerkin method for numerical solution of conservation laws and convection – diffusion problems, (), 7-32
[8] G. Chiocchia, Exact solutions to transonic and superesonic flows, Technical Report AGARD-AR-211, Center for Aerospace Information, NASA, 1985
[9] Cockburn, B., Discontinuous Galerkin methods for convection dominated problems, (), 69-224 · Zbl 0937.76049
[10] Cockburn, B.; Hou, S.; Shu, C.W., TVB runge – kutta local projection discontinuous Galerkin finite element for scalar conservation laws II: general framework, Math. comp., 52, 411-435, (1989) · Zbl 0662.65083
[11] ()
[12] Dick, E., Second-order formulation of a multigrid method for steady Euler equations through defect-correction, J. comput. appl. math., 35, 1-3, 159-168, (1991) · Zbl 0724.76059
[13] Dolejšı́, V., Anisotropic mesh adaptation technique for viscous flow simulation, East-west J. numer. math., 9, 1, 1-24, (2001) · Zbl 1056.76045
[14] Dolejšı́, V., A higher order scheme based on the finite volume approach, (), 333-340 · Zbl 1059.65510
[15] Dolejšı́, V.; Feistauer, M., On the discontinuous Galerkin method for the numerical solution of compressible high-speed flow, (), 65-84 · Zbl 1276.76039
[16] Dolejšı́, V.; Feistauer, M.; Schwab, C., A finite volume discontinuous Galerkin scheme for nonlinear convection-diffusion problems, Calcolo, 39, 1-40, (2002) · Zbl 1098.65095
[17] Dolejšı́, V.; Feistauer, M.; Schwab, C., On discontinuous Galerkin methods for nonlinear convection – diffusion problems and compressible flow, Mathematica bohemica, 127, 2, 163-179, (2002) · Zbl 1074.65522
[18] Dolejšı́, V.; Feistauer, M.; Schwab, C., On some aspects of the discontinuous Galerkin finite element method for conservation laws, Math. comput. simul., 61, 333-346, (2003) · Zbl 1013.65108
[19] Eymard, R.; Gallouët, T.; Herbin, R., Solution of equations in Rn (part 3). techniques of scientific computing (part 3), Handbook of numerical analysis, vol. t, (2000), North-Holland/Elsevier Amsterdam, Finite volume methods (chapter), p. 713-1-20
[20] Feistauer, M., Mathematical methods in fluid dynamics, (1993), Longman Scientific & Technical Harlow · Zbl 0819.76001
[21] Feistauer, M., Discontinuous Galerkin method: compromise between FV and FE schemes, (), 81-95 · Zbl 1118.65365
[22] Feistauer, M.; Felcman, J., Theory and applications of numerical schemes for nonlinear convection-diffusion problems and compressible navier – stokes equations, (), 175-194 · Zbl 0891.76051
[23] Feistauer, M.; Felcman, J.; Dolejšı́, V., Numerical simulation of compresssible viscous flow through cascades of profiles, Zamm, 76, S4, 297-300, (1996) · Zbl 0925.76443
[24] Feistauer, M.; Felcman, J.; Straškraba, I., Mathematical and computational methods for compressible flow, (2003), Clarendon Press Oxford · Zbl 1028.76001
[25] Felcman, J.; Šolı́n, P., On the construction of the osher – solomon scheme for 3D Euler equations, East-west J. numer. math., 6, 1, 43-64, (1998) · Zbl 0912.76052
[26] Fezoui, L.; Stoufflet, B., A class of implicit upwind schemes for Euler simulations with unstructured meshes, J. comput. phys., 84, 1, 174-206, (1989) · Zbl 0677.76062
[27] Hartmann, R.; Houston, P., Adaptive discontinuous Galerkin finite element methods for nonlinear hyperbolic conservation laws, SIAM J. sci. comp., 24, 979-1004, (2002) · Zbl 1034.65081
[28] Hemker, P.W.; Spekreijse, S.P., Multiple grid and osher’s scheme for the efficient solution of the steady Euler equations, Appl. numer. math., 2, 475-493, (1986) · Zbl 0612.76077
[29] Hirsch, C., Numerical computation of internal and external flows, ()
[30] Koren, B.; Hemker, P.W., Damped, direction-dependent multigrid for hypersonic flow computations, Appl. numer. math., 7, 4, 309-328, (1991) · Zbl 0733.76033
[31] Kröner, D., Numerical schemes for conservation laws, (1997), Wiley Teubner Stuttgart · Zbl 0872.76001
[32] Meister, A., Comparison of different Krylov subspace methods embedded in an implicit finite volume scheme for the computation of viscous and inviscid flow fields on unstructured grids, J. comput. phys., 140, 311-345, (1998) · Zbl 0935.76051
[33] Osher, S.; Solomon, F., Upwind difference schemes for hyperbolic systems of conservation laws, Math. comp., 38, 339-374, (1982) · Zbl 0483.65055
[34] Saad, Y.; Schultz, M.H., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. sci. stat. comput., 7, 856-869, (1986) · Zbl 0599.65018
[35] Shu, C.W., Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, (), 325-432 · Zbl 0927.65111
[36] Spekreijse, S.P., Multigrid solution of the steady Euler equations, (1988), Centre for Mathematics and Computer Science Amsterdam · Zbl 0643.76068
[37] B. Stoufflet, Implicit finite element methods for the Euler equations. In Numerical methods for the Euler equations of fluid dynamics, Proc. INRIA Workshop, Rocquencourt/France 1983, 1985, pp. 409-434
[38] Toro, E.F., Riemann solvers and numerical methods for fluid dynamics, (1997), Springer Berlin · Zbl 0888.76001
[39] van der Vegt, J.J.W.; van der Ven, H., Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flow, J. comput. phys., 182, 546-585, (2002) · Zbl 1057.76553
[40] vander Ven, H.; vander Vegt, J.J.W., Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows. II. efficient flux quadrature, Comput. methods appl. mech. engrg., 191, 4747-4780, (2002) · Zbl 1099.76521
[41] Vijayasundaram, G., Transonic flow simulation using upstream centered scheme of Godunov type in finite elements, J. comput. phys., 63, 416-433, (1986) · Zbl 0592.76081
[42] Wesseling, P., Principles of computational fluid dynamics, (2001), Springer Berlin · Zbl 0989.76069
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