Abstract
To solve flow problems associated with the Navier-Stokes equations, we construct a mixed finite volume/finite element method for the spatial approximation of the convective and diffusive parts of the flux, respectively. The finite volume component of the method is adapted from the authors’ construction ([1], [2], [3]), for hyperbolic conservation laws and unstructured triangular or rectangular grids, of 2-dimensional finite volume extensions of the Lax-Friedrichs and Nessyahu-Tadmor central difference schemes, in which the resolution of Riemann problems at cell interfaces is by-passed thanks to the use of the Lax-Friedrichs scheme on two specific staggered grids. Piecewise linear cell interpolants, slope limiters and a 2-step time discretization lead to an oscillation-free second order resolution.
For the viscous terms we use a centred finite element approximation inspired by [9], [11].
Numerical experiments on classical test problems including comparison with other methods lead to fairly competitive results with favourable computing times and sharper shock capture.
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References
P. Arminjon and M.C. Viallon, Géénralisation du schéma de Nessyahu-Tadmor pourune équation hyperbolique à deux dimensions d’espace, Comptes Rendus de l’Acad. des Sciences, Paris, t. 320, série I, 85–88, January 1995.
P. Arminjon, M.C. Viallon and A. Madrane, A Finite Volume Extension of the LaxFriedrichs and Nessyahu-Tadmor Schemes for Conservation Laws on Unstructured Grids, Int.J. Comp.Fluid Dynamics, 9 (1997), 1–22.
P.Arminjon, D.Stanescu and M.C. Viallon, A two-dimensional finite volume extension of the Lax-Friedrichs and Nessyahu-Tadmor schemes for compressible flows, Proc. of the 6 th. Int. Symp. on Comp. Fluid Dynamics, Lake Tahoe (Nevada), September 4–8, 1995, M. Hafez and K.Oshima, editors, 4, 7–14.
P. Arminjon, A. Madrane and M.C. Viallon, Comparison of a finite volume version of the Lax-Friedrichs and Nessyahu-Tadmor schemes and discontinuous finite element methods for compressible flows on unstructured grids, Proceedings of a Symposium Honoring S.K.Godunov, May 1–2, 1997, The University of Michigan, Ann Arbor (Mi.), B. van Leer, editor, to appear as a special volume in J. Comp. Physics.
P. Arminjon, M.C. Viallon, A. Madrane and H. Kaddouri, Discontinuous finite elements and a 2-dimensional finite volume generalization of the Lax-Friedrichs and Nessyahu-Tadmor schemes for compressible flows on unstructured grids, to appear in C.F.D. Review, M. Hafez and K. Oshima, editors, (1997).
M.C. Viallon and P. Arminjon, Convergence of a finite volume extension of the Nessyahu-Tadmor scheme on unstructured grids for a 2-dimensional linear hyperbolic equation, Res. Rep. No. 2239, CRM, Univ-de Montréal, January 1995, to appear in SIAM J. Num. Anal. For the nonlinear case: to appear in Num. Mathematik, (1995).
R. Sanders and A. Weiser, A high order staggered mesh approach for nonlinear hyperbolic system of conservation laws, JCP 1010, (1992), 314–329. 739CP, The 12th AIAA CFD Conf., 1995.
R. Peyret and T.D. Taylor, Computational Methods for Fluid Flow, Springer-Ver lag, New-York, Heidelberg, Berlin, (1983).
P. Rostand and B. Stoufflet, Finite Volume Galerkin Methods for Viscous Gas Dynamics, INRIA Res. Rep. No. 863, Rocquencourt, 78153 Le Chesnay, France, (1988).
H. Nessyahu and E. Tadmor, Non-oscillatory Central Differencing for Hyperbolic Conservation Laws, J. Comp. Physics 87, 2 (1990), 408–463.
B. Palmerio, L.Fezoui, C. Olivier and A. Dervieux, On TVD criteria for mesh adaption for Euler and Navier-Stokes calculations, INRIA Res. Rep. No. 1175, Rocquencourt, 78153 Le Chesnay, France, (1990)
J. Jaffré and L. Kaddouri, Discontinuous finite elements for the Euler equations. Proc. 3d. Int. Conf. on Hyperbolic Problems, June 11–15, 1990, Uppsala (Sweden). B. Engquist and B. Gustafsson, editors, Studentlitteratur, Chartwell-Bratt, 2 (1991). 602–610.
B. van Leer, Towards the Ultimate Conservative Difference Scheme. II. Monotonicity and conservation combined in a second order scheme, J. Comp. Phys., 14 (1974) 361–370.
S. Champier, Convergence de schémas numériques type Volumes Finis pour k résolution d’équations hyperboliques, Thése, Univ. de St-Étienne, 1992.
J. Steger, R.F. Warming, Flux vector splitting for the inviscid gas dynamic with applications to finite-difference methods, J. of Comp. Phys., 40(2) (1981), 263–293.
M.J. Castro Diaz and F. Hecht, Anisotropie Surface Mesh Generation, INRIA Res. Rep. No. 2672, October 1995, INRIA, Rocquencourt, 78153 Le Chesnay, France.
B.V.K. Satya Sai, O.C. Zienkiewicz, M.T. Manzari, P.R.M. Lyra and K. Morgan. General purpose versus special algorithms for high-speed flows with shocks, Int. J. Num. Meth. in Fluids, 27 (1998), 57–80.
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Arminjon, P., Madrane, A. (1999). A Mixed Finite Volume/Finite Element Method for 2-dimensional Compressible Navier-Stokes Equations on Unstructured Grids. In: Fey, M., Jeltsch, R. (eds) Hyperbolic Problems: Theory, Numerics, Applications. International Series of Numerical Mathematics, vol 129. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8720-5_2
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