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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.

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
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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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