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Finite volume solvers and moving least-squares approximations for the compressible Navier-Stokes equations on unstructured grids. (English) Zbl 1173.76358
Summary: This paper explores the approximation power of Moving Least-Squares (MLS) approximations in the context of higher-order finite volume schemes on unstructured grids. The scope of the application of MLS is threefold: (1) computation of high-order derivatives of the field variables for a Godunov-type approach to hyperbolic problems or terms of hyperbolic character, (2) direct reconstruction of the fluxes at cell edges, for elliptic problems or terms of elliptic character, and (3) multiresolution shock detection and selective limiting. A major advantage of the proposed methodology over the most popular existing higher-order methods is related to the viscous discretization. The use of MLS approximations allows the direct reconstruction of high-order viscous fluxes using quite compact stencils, and without introducing new degrees of freedom, which results in a significant reduction in storage and workload. A selective limiting procedure is proposed, based on the multiresolution properties of the MLS approximants, which allows to switch off the limiters in smooth regions of the flow. Accuracy tests show that the proposed method achieves the expected convergence rates. Representative simulations show that the methodology is applicable to problems of engineering interest.

MSC:
76M12Finite volume methods (fluid mechanics)
76N15Gas dynamics, general
Software:
HLLE
WorldCat.org
Full Text: DOI
References:
[1] Cockburn, B.; Shu, C. W.: Runge -- Kutta discontinuous Galerkin methods for convection-dominated problems. J. scient. Comput. 6, 173-261 (2001) · Zbl 1065.76135
[2] Wang, Z. J.; Zhang, L.; Liu, Y.: Spectral (finite) volume method for conservation laws on unstructured grids IV: Extension to two-dimensional Euler equations. J. comput. Phys. 194, 716-741 (2004) · Zbl 1039.65072
[3] Lancaster, P.; Salkauskas, K.: Surfaces generated by moving least squares methods. Math. comput. 155, 141-158 (1981) · Zbl 0469.41005
[4] Liu, W. K.; Li, S.; Belytschko, T.: Moving least-squares reproducing kernel methods: (I) methodology and convergence. Comput. methods appl. Mech. engrg. 143, 113 (1997) · Zbl 0883.65088
[5] Li, S.; Liu, W. K.: Moving least-squares reproducing kernel methods: (II) Fourier analysis. Comput. methods appl. Mech. engrg. 139, 159-193 (1996) · Zbl 0883.65089
[6] Li, S.; Liu, W. K.: Reproducing kernel hierarchical partition of unity, part I -- formulation and theory. Int. J. Numer. methods engrg. 45, 251-288 (1999) · Zbl 0945.74079
[7] A. Gossler, Moving Least-Squares: a numerical differentiation method for irregularly spaced calculation points, SANDIA Report, SAND2001-1669, 2001.
[8] Levin, D.: The approximation power of moving least-squares. Math. comput. 67, No. 224, 1517-1531 (1998) · Zbl 0911.41016
[9] Cueto-Felgueroso, L.; Colominas, I.; Fe, J.; Navarrina, F.; Casteleiro, M.: High order finite volume schemes on unstructured grids using moving least-squares reconstruction. Application to shallow water dynamics. Int. J. Numer. methods engrg. 65, 295-331 (2006) · Zbl 1111.76032
[10] Godunov, S. K.: A difference method for the numerical calculation of discontinuous solutions of hydrodynamic equations. Mat. sbornik. 47, No. 3, 271-306 (1959)
[11] Harten, A.; Lax, P.; Van Leer, B.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM rev. 25, 35-61 (1983) · Zbl 0565.65051
[12] T.J. Barth, D.C. Jespersen, The design and application of upwind schemes on unstructured meshes, AIAA-89-0366, 1989.
[13] Frink, N. T.: Upwind scheme for solving the Euler equations on unstructured tetrahedral meshes. Aiaa j. 30, No. 1, 70 (1992) · Zbl 0741.76043
[14] Venkatakrishnan, V.: Convergence to steady state solutions of the Euler equations on unstructured grids with limiters. J. comput. Phys. 118, 120-130 (1995) · Zbl 0858.76058
[15] Jawahar, P.; Kemath, H.: A high-resolution procedure for Euler and Navier -- Stokes computations on unstructured grids. J. comput. Phys. 164, 165-203 (2000) · Zbl 0992.76063
[16] T.J. Barth, Aspects of unstructured grids and finite-volume solvers for the Euler and Navier -- Stokes equations, VKI Lecture Series 1994-05, 1995.
[17] J.M. Vaassen, P. Wautelet, J.A. Essers, A quadratic reconstruction scheme for hypersonic reacting flows on unstructured meshes, in: ECCOMAS Computational Fluid Dynamics Conference, Swansea, Wales. · Zbl 1057.76043
[18] Aavatsmark, I.: An introduction to multipoint flux approximations for quadrilateral grids. Comput. geosci. 6, 405-432 (2002) · Zbl 1094.76550
[19] Liu, W. K.; Hao, W.; Chen, Y.; Jun, S.; Gosz, J.: Multiresolution reproducing kernel particle methods. Comput. mech. 20, 295-309 (1997) · Zbl 0893.73078
[20] Sjögreen, B.; Yee, H. C.: Multiresolution wavelet based adaptive numerical dissipation control for high order methods. J. scient. Comput. 20, 211-255 (2004) · Zbl 1106.76411
[21] White, F. M.: Viscous fluid flow. (1991)
[22] L. Cueto-Felgueroso, I. Colominas, High-order finite volume methods and multiresolution reproducing kernels, Arch. Comput. Methods Engrg., submitted for publication. · Zbl 1300.76018
[23] L. Cueto-Felgueroso, Particles, finite volumes and unstructured grids: numerical simulation of fluid dynamics problems, Ph.D. thesis, Universidad de A Coruña, 2005. <http://www.tesisenred.net/> (in Spanish).
[24] Cueto-Felgueroso, L.; Colominas, I.; Mosqueira, G.; Navarrina, F.; Casteleiro, M.: On the Galerkin formulation of the SPH method. Int. J. Numer. methods engrg. 60, 1475-1512 (2004) · Zbl 1060.76638
[25] Van Leer, B.: Towards the ultimate conservative difference scheme V. A second order sequel to Godunov’s method. J. comput. Phys. 32, 101 (1979)
[26] Van Rosendale, J.: Floating shock Fitting via Lagrangian adaptive meshes. Icase 94-89 (1989)
[27] Van Albada, G. D.; Van Leer, B.; Roberts, W. W.: A comparative study of computational methods in cosmic gas dynamics. Astron. astrophys. 108, 76 (1982) · Zbl 0492.76117
[28] Roe, P. L.: Approximate Riemann solvers, parameter vectors and difference schemes. J. comput. Phys. 43, 357-372 (1981) · Zbl 0474.65066
[29] Shu, C. -W.; Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. comput. Phys. 77, 439-471 (1988) · Zbl 0653.65072
[30] Shapiro, A. H.: The dynamics and thermodynamics of compressible fluid flow. (1953) · Zbl 0053.00803
[31] Shu, C. W.; Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes, II. J. comput. Phys. 83, 32-78 (1989) · Zbl 0674.65061
[32] Yee, H. C.; Sandham, N. D.; Djomehri, M. J.: Low-dissipative high-order shock-capturing methods using characteristic-based filters. J. comput. Phys. 150, 99-238 (1999) · Zbl 0936.76060