zbMATH — the first resource for mathematics

On a robust discontinuous Galerkin technique for the solution of compressible flow. (English) Zbl 1114.76042
Summary: We describe a numerical technique allowing the solution of compressible inviscid flow with a wide range of Mach numbers. The method is based on the application of discontinuous Galerkin finite element method for the space discretization of Euler equations written in conservative form, combined with a semi-implicit time discretization. Special attention is paid to the treatment of boundary conditions and to the stabilization of the method in the vicinity of discontinuities avoiding Gibbs phenomenon. As a result, we obtain a technique allowing the numerical solution of flows with practically all Mach numbers without any modification of Euler equations. This means that the proposed method can be used for solution of high-speed flows as well as low Mach number flows. Presented numerical tests prove the accuracy of the method and its robustness with respect to Mach number.

76M10 Finite element methods applied to problems in fluid mechanics
76N15 Gas dynamics (general theory)
Full Text: DOI
[1] 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
[2] Baumann, C.E.; Oden, J.T., A discontinuous hp finite element method for the Euler and navier – stokes equations, Int. J. numer. methods fluids, 31, 79-95, (1999) · Zbl 0985.76048
[3] ()
[4] Darwish, M.; Moukalled, F.; Sekar, B., A robust multi-grid pressure-based algorithm for multi-fluid flow at all speeds, Int. J. numer. methods fluids, 41, 1221-1251, (2003) · Zbl 1047.76057
[5] Davis, T.A.; Duff, I.S., A combined unifrontal/multifrontal method for unsymmetric sparse matrices, ACM trans. math. software, 25, 1-19, (1999) · Zbl 0962.65027
[6] Dolejšı´, V., Anisotropic mesh adaptation for finite volume and finite element methods on triangular meshes, Comput. vis. sci., 1, 3, 165-178, (1998) · Zbl 0917.68214
[7] Dolejšı´, V.; Feistauer, M., On the discontinuous Galerkin method for the numerical solution of compressible high-speed flow, (), 65-84 · Zbl 1276.76039
[8] Dolejšı´, V.; Feistauer, M., A semi-implicit discontinuous Galerkin finite element method for the numerical solution of inviscid compressible flow, J. comput. phys., 198, 727-746, (2004) · Zbl 1116.76386
[9] V. Dolejšı´, M. Feistauer, J. Hozman, Analysis of semi-implicit DGFEM for nonlinear convection – diffusion problems on nonconforming meshes. Comput. Methods Appl. Mech. Eng. (in press), doi:10.1016/j.cma.2006.09.025. · Zbl 1121.76033
[10] Dolejšı´, 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
[11] Feistauer, M., Mathematical methods in fluid dynamics, (1993), Longman Scientific & Technical Harlow · Zbl 0819.76001
[12] Feistauer, M.; Felcman, J.; Straškraba, I., Mathematical and computational methods for compressible flow, (2003), Clarendon Press Oxford · Zbl 1028.76001
[13] Feistauer, M.; Švadlenka, K., Discontinuous Galerkin method of lines for solving nonstationary singularly perturbed linear problems, J. numer. math., 2, 97-117, (2004) · Zbl 1059.65083
[14] Fraenkel, L., On corner eddies in plane inviscid shear flow, J. fluid mech., 11, 400-406, (1961) · Zbl 0113.40602
[15] Hauke, G.; Hughes, T.J.R., A comparative study of different sets of variables for solving compressible and incompressible flows, Comput. methods appl. mech. eng., 153, 1-44, (1998) · Zbl 0957.76028
[16] Jaffre, J.; Johnson, C.; Szepessy, A., Convergence of the discontinuous Galerkin finite elements method for hyperbolic conservation laws, Math. models methods appl. sci., 5, 367-386, (1995) · Zbl 0834.65089
[17] Klein, R., Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics 1: one-dimensional flow, J. comput. phys., 121, 213-237, (1995) · Zbl 0842.76053
[18] Meister, A., Viscous flow fields at all speeds: analysis and numerical simulation, J. appl. math. phys., 54, 1010-1049, (2003) · Zbl 1141.76441
[19] Meister, A.; Struckmeier, J., Hyperbolic partial differential equations, theory, numerics and applications, (2002), Vieweg Braunschweig
[20] Park, J.H.; Munz, C.-D., Multiple pressure variables methods for fluid flow at all Mach numbers, Int. J. numer. methods fluids, 49, 905-931, (2005) · Zbl 1170.76342
[21] Roller, S.; Munz, C.-D.; Geratz, K.J.; Klein, R., The multiple pressure variables method for weakly compressible fluids, Z. angew. math. mech., 77, 481-484, (1997) · Zbl 0900.76515
[22] van der Heul, D.R.; Vuik, C.; Wesseling, P., A conservative pressure-correction method for flow at all speeds, Comput. fluids, 32, 1113-1132, (2003) · Zbl 1046.76033
[23] 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
[24] 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
[25] Wenneker, I.; Segal, A.; Wesseling, P., A Mach-uniform unstructured staggered grid method, Int. J. numer. methods fluids, 40, 1209-1235, (2002) · Zbl 1025.76023
[26] Wesseling, P., Principles of computational fluid dynamics, (2001), Springer Berlin · Zbl 0989.76069
[27] Wesseling, P.; van der Heul, D.R., Uniformly effective numerical methods for hyperbolic systems, Computing, 66, 249-267, (2001) · Zbl 1017.76056
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.