Positivity statements for a mixed-element-volume scheme on xed and moving grids
Keywords:
nite-element, nite-volume, positivity, maximum principle, scalar conservation law, Euler, ALE, multi-component, uid-structure interactionAbstract
This paper considers a class of second-order accurate vertex-centered mixed niteelement nite-volume MUSCL schemes. These schemes apply to unstructured triangulations and tetrahedrizations and uxes are computed on an edge basis. We dene conditions under which these schemes satisfy a density-positivity statement for Euler ows, a maximum principle for a scalar conservation law and a multicomponent ow. This extends to an Arbitrary- Lagrangian-Eulerian formulation. Steady and unsteady ow simulations illustrate the accuracy and the robustness of these schemes.
Downloads
References
Abgrall R., On essentially non-oscillatory schemes on unstructured meshes, analysis and implementation
, Journal of Computational Physics, vol. 114, p. 45-58, 1994.
Abgrall R., How to prevent pressure oscillations in multicomponent ows: a quasi conservative
approach, Journal of Computational Physics, vol. 125, p. 150-160, 1996.
Arminjon P., Dervieux A., Construction of TVD-like articial viscosities on two-dimensional
arbitrary FEM grids, Journal of Computational Physics, vol. 106, p. 106, No 1, 176-198,
Arminjon P., Viallon M., Convergence of a nite volume extension of the Nessyahu-Tadmor
scheme on unstructured grids for a two-dimensional linear hyperbolic equation, SIAM J.
Num. Analysis, vol. 36, n 3, p. 738-771, 1999.
Baba K., Tabata M., On a conservative upwind nite element scheme for convective diffusion
equations, R.A.I.R.O Numer. Anal., vol. 15, p. 3-25, 1981.
Camarri S., Koobus B., Salvetti M.-V., Dervieux A., A low-diffusion MUSCL scheme for
LES on unstructured grids, Computers and Fluids, vol. 33, p. 1101-1129, 2004.
Catalano L. A., A new reconstruction scheme for the computation of inviscid compressible
ows on 3D unstructured grids, International Journal for Numerical Methods in Fluids,
vol. 40, n 1-2, p. 273-279, 2002.
Cockburn B., Shu C.-W., TVB Runge-Kutta local projection discontinuous Galerkin nite element
method for conservation laws II: General framework, Mathematics of Computation,
vol. 52, p. 411-435, 1989.
Debiez C., Approximation et linéarisation d'écoulements aérodynamiques instationnaires, PhD
thesis, University of Nice, France (in French), 1996.
Debiez C., Dervieux A., Mer K., Nkonga B., Computation of Unsteady Flows with Mixed Finite
Volume/Finite Element Upwind Methods, International Journal for Numerical Methods
in Fluids, vol. 27, p. 193-206, 1998.
Deconinck H., Roe P., Struijs R., A Multidimensional Generalization of Roe's Flux Difference
Splitter for the Euler Equations, Computers and Fluids, vol. 22, n 23, p. 215-222, 1993.
Einfeldt B., Munz C. D., Roe P. L., Sjogreen B., On Godunov-type methods near low densities
, Journal of Computational Physics, vol. 92, p. 273-295, 1991.
Farhat C., High performance simulation of coupled nonlinear transient aeroelastic problems,
Special course on parallel computing in CFD. n R-807, NATO AGARD Report, October,
Farhat C., Geuzaine P., Grandmont C., The Discrete Geometric Conservation Law and the
nonlinear stability of ALE schemes for the solution of ow problems on moving grids, J.
of Computational Physics, vol. 174, p. 669-694, 2001.
Farhat C., Lesoinne M., Two efcient staggered procedures for the serial and parallel solution
of three-dimensional nonlinear transient aeroelastic problems, Comput. Meths. Appl.
Mech. Engrg., vol. 182, p. 499-516, 2000.
Fezoui L., Dervieux A., Finite-element non oscillatory schemes for compressible ows, Symposium
on Computational Mathematics and Applications n 730, Publications of university
of Pavie (Italy), 1989a.
Fezoui L., Stoufet B., A class of implicit schemes for Euler simulations with unstructured
meshes, Journal of Computational Physics, vol. 84, n 1, p. 174-206, 1989b.
Godlewski E., Raviart P.-A., Numerical approximation of hyperbolic systems of conservation
law, Springer, 1996.
Harten A., High resolution schemes for hyperbolic conservation laws, Journal of Computational
Physics, vol. 49, p. 357-393, 1983.
Harten A., Engquist B., Osher S., Chakravarthy S., Uniformly high-order accurate essentially
non oscillatory schemes, III, Journal of Computational Physics, vol. 71, p. 231-303, 1987.
Jameson A., Articial Diffusion, Upwind Biasing, Limiters and their Effect on Accuracy and
Multigrid Convergence in Transonic and Hypersonic Flows, AIAA paper 93-3359, 1993.
Larrouturou B., How to Preserve the Mass Fraction Positivity when Computing Compressible
Multi-component Flows, Journal of Computational Physics, vol. 95, p. 59-84, 1991.
Linde T., Roe P., On multidimensional positively conservative high resolution schemes, AIAA
paper 97-2098, 1998.
Mer K., Variational analysis of a mixed element/volume scheme with fourth-order viscosity
on general triangulations, Computer Methods in Applied Mechanics and Engineering, vol.
, p. 45-62, 1998.
Murrone A., Guillard H., A ve equation model for compressible two-phase ow computations
, Journal of Computational Physics, vol. 202, n 2, p. 664-698, 2005.
Perthame B., Khobalate B., Maximum Principle on the Entropy and Minimal Limitations for
Kinetic Scheme, Rapport de recherche n 1628, INRIA, 1992.
Perthame B., Shu C., On positivity preserving nite-volume schemes for Euler equations,
Numerical Mathematics, vol. 73, p. 119-130, 1996.
Piperno S., Depeyre S., Criteria for the Design of Limiters yielding efcient high resolution
TVD schemes, Comput. & Fluids, vol. 27, n 2, p. 183-197, 1998.
Selmin V., Formaggia L., Unied construction of nite element and nite volume discretizations
for compressible ows, International Journal for Numerical Methods in Engineering,
vol. 39, n 1, p. 1-32, 1998.
Shu C., Osher S., Efcient Implementation of Essential Non-oscillatory Shock Capturing
Schemes, Journal of Computational Physics, vol. 77, p. 439-471, 1988.
Sidilkover D., A Genuinely Multidimensional Upwind Scheme and Efcient Multigrid Solver
for the Compressible Euler Equations, ICASE report no. 94-84, 1994.
Stoufet B., Periaux J., Fezoui L., Dervieux A., 3-D Hypersonic Euler Numerical Simulation
around Space Vehicles using Adapted Finite Elements, 25th AIAA Aerospace Meeting,
Reno (1987), AIAA paper 86-0560, 1996.
Sweby P. K., High resolution schemes using limiters for hyperbolic conservation laws, SIAM
Journal in Numerical Analysis, vol. 21, p. 995-1011, 1984.
Van Leer B., Towards the Ultimate Conservative Difference Scheme V: A Second Order
Sequel to Godunov's Method, Journal of Computational Physics, vol. 32, p. 101-136,
Venkatakrishnan V., A perspective on unstructured grid ow solvers, AIAA Journal, vol. 34,
p. 533-547, 1996.
Venkatakrisnan V. et al., Barriers and Challenges in CFD, ICASE Workshop, Kluwer Acad.
Pub., ICASE/LaRC Interdisciplinary Series, vol. 6, p. 299-313, 1998.
Whitaker D., Grossman B., Löhner R., Two-Dimensional Euler Computations on a Triangular
Mesh Using an Upwind, Finite-Volume Scheme, AIAA paper 89-0365, 1989.
Yates E., AGARD standard aeroelastic conguration for dynamic response, candidate conguration
I. - Wing 445.6, NASA TM-100492, 1987.
