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High-order ENO and WENO schemes for unstructured grids. (English) Zbl 1388.76217
Summary: This work describes the implementation and analysis of high-order accurate schemes applied to high-speed flows on unstructured grids. The class of essentially non-oscillatory schemes (ENO), that includes weighted ENO schemes (WENO), is discussed in the paper with regard to the implementation of third- and fourth-order accurate methods. The entire reconstruction process of ENO and WENO schemes is described with emphasis on the stencil selection algorithms. The stencils can be composed by control volumes with any number of edges, e.g. triangles, quadrilaterals and hybrid meshes. In the paper, ENO and WENO schemes are implemented for the solution of the dimensionless, 2-D Euler equations in a cell centred finite volume context. High-order flux integration is achieved using Gaussian quadratures. An approximate Riemann solver is used to evaluate the fluxes on the interfaces of the control volumes and a TVD Runge-Kutta scheme provides the time integration of the equations. Such a coupling of all these numerical tools, together with the high-order interpolation of primitive variables provided by ENO and WENO schemes, leads to the desired order of accuracy expected in the solutions. An adaptive mesh refinement technique provides better resolution in regions with strong flowfield gradients. Results for high-speed flow simulations are presented with the objective of assessing the implemented capability.

MSC:
76M20 Finite difference methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
Software:
AUSM
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[1] . High-order unstructured grid ENO and WENO schemes applied to aerodynamics flows. Proceedings of the 17th AIAA Computational Fluid Dynamics Conference, AIAA Paper No. 2005-5115, Toronto, Canada, 2005.
[2] . Comparison of unstructured grid finite volume methods for cold gas hypersonic flow simulations. Proceedings of the 16th AIAA Applied Aerodynamics Conference, AIAA Paper No. 98-2629, Albuquerque, New Mexico, 1998.
[3] Figueira da Silva, Journal of Computational Physics 160 pp 522– (2000)
[4] Azevedo, Journal of the Brazilian Society of Mechanical Sciences and Engineering (2007)
[5] Flux–vector splitting for the Euler equations. Proceedings of the 8th International Conference on Numerical Methods in Fluid Dynamics. Lecture Notes in Physics, vol. 170. Springer: Berlin, 1982; 507–512.
[6] Liou, Journal of Computational Physics 129 pp 364– (1996)
[7] Roe, Journal of Computational Physics 43 pp 200– (1981)
[8] Anderson, AIAA Journal 24 pp 1453– (1986)
[9] Numerical Computation of Internal and External Flows, vol. 2. Wiley: New York, 1990.
[10] Harten, Journal of Computational Physics 71 pp 231– (1987)
[11] Liu, Journal of Computational Physics 115 pp 200– (1994)
[12] Sonar, Computer Methods in Applied Mechanics and Engineering 140 pp 157– (1997)
[13] Friedrich, Journal of Computational Physics 144 pp 194– (1998)
[14] Shu, Journal of Computational Physics 77 pp 439– (1988)
[15] High order ENO schemes for unstructured meshes based on least-squares reconstruction. Report No. P631-1296, Argonne National Laboratory, Mathematics and Computer Science Division, 1997.
[16] Abgrall, Journal of Computational Physics 114 pp 45– (1994)
[17] Hu, Journal of Computational Physics 150 pp 97– (1999)
[18] Jiang, Journal of Computational Physics 126 pp 77– (1996)
[19] . Multi-dimensional ENO schemes for general geometries. ICASE Report No. 91-76, 1991.
[20] The Dynamics and Thermodynamics of Compressible Fluid Flow, vol. 2. The Ronald Press: New York, 1954.
[21] . Higher order upwind finite volume schemes with ENO-properties for general unstructured meshes. AGARD Report No. 787, 1992.
[22] Woodward, Journal of Computational Physics 54 pp 115– (1984)
[23] . Essentially non-oscillatory schemes on cold gas hypersonic flow simulations. Proceedings of the XXVI Iberian Latin–American Congress on Computational Methods in Engineering, Guarapari, ES, Brazil, 2005.
[24] . Blunt body flow simulations. 24th AIAA Joint Propulsion Conference, AIAA Paper No. 88-2904, Boston, MA, 1988.
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