Discrete conservation properties of unstructured mesh schemes

Annual Review of Fluid Mechanics

Volumn 43, Issue , 2011, Pages 299-318

  Perot, J Blair ^a  

^a University of Massachusetts   (United States)

Author keywords

computational fluid dynamics; discrete calculus methods; global and local conservation; primary and secondary conservation

Indexed keywords

CONSERVATION PROPERTIES; DISCRETE CALCULUS METHODS; LOCAL CONSERVATION; NUMERICAL SOLUTION; PHYSICAL CONSTRAINTS; PRIMARY AND SECONDARY CONSERVATION; TECHNIQUES USED; UNSTRUCTURED MESHES;

CALCULATIONS; FLUID DYNAMICS; FLUIDS; NUMERICAL METHODS;

COMPUTATIONAL FLUID DYNAMICS;

COMPUTATIONAL FLUID DYNAMICS; ENTROPY; KINETIC ENERGY; LITERATURE REVIEW; NUMERICAL MODEL; VORTICITY;

EID: 79951967877     PISSN: 00664189     EISSN: None     Source Type: Book Series    
DOI: 10.1146/annurev-fluid-122109-160645     Document Type: Article

References (77)

1. Ascher UM, McLachlan RI. 2005. On symplectic and multisymplectic schemes for the KdV equation. J. Sci. Comput. 25(1):83-104 (Pubitemid 41766587)
2. Benhamadouche S, Laurence D. 2002. Global kinetic energy conservation with unstructured meshes. Int. J. Numer. Methods Fluids 40:561-71
3. Boffi D, Fernandes P, Gastaldi L, Perugia I. 1999. Computational models of electromagnetic resonators: analysis of edge element approximation. SIAM J. Numer. Anal. 36:1264-90 (Pubitemid 129567512)
4. Bossavit A, Mayergoyz I. 1989. Edge elements for scattering problems. IEEE Trans. Mag. 25(4):2816-21
5. Bradshaw P, Cebeci T, Whitelaw JH. 1981. Engineering Calculation Methods for Turbulent Flow. New York: Academic
6. Bridges TJ, Reich S. 2001. Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity. Phys. Lett. A 284:184-93 (Pubitemid 32566397)
7. Cebeci T, Chang KC, Bradshaw P. 1980. Solution of a hyperbolic system of turbulence-model equations by the "box" scheme. Comput. Methods Appl. Mech. Eng. 22:213-27 (Pubitemid 11421514)
8. Chang W, Giraldo F, Perot JB. 2002. Analysis of an exact fractional step method. J. Comput. Phys. 179:1-17
9. Chattot J-J. 1999. A conservative box scheme for the Euler equations. Int. J. Numer. Methods Fluids 31:149-58
10. Costabel M, Dauge M. 2003. Computation of resonance frequencies for Maxwell equations in non smooth domains. In Lecture Notes in Computational Science and Engineering, ed. M Ainsworth, P Davies, DB Duncan, PA Martin, B Rynne, pp. 125-62. Top. Comput. Wave Propag. Vol. 31. New York: Springer
11. Coyle J, Ledger P. 2003. Evidence of exponential convergence in the computation of Maxwell eigenvalues. Comput. Methods Appl. Mech. Eng. 194:587-604 (Pubitemid 39637503)
12. Croisille J-P, Greff I. 2005. An efficient box-scheme for convection-diffusion equations with sharp contrast in the diffusion coefficients. Comput. Fluids 34:461-89 (Pubitemid 40032120)
13. Dai M, Wang H, Perot JB, Schmidt DP. 2002. Direct interface tracking of droplet deformation. At. Sprays 12:721-35
14. De Rham G. 1931. Sur l'analysis situs des varietes a n dimensions. J. Math. 10:115-99
15. Feiereisen WJ, Reynolds WC, Ferziger JH. 1981. Numerical simulation of a compressible, homogeneous, turbulent shear flow. Rep. TF-13, Thermosci. Div., Mech. Eng., Stanford Univ.
16. Felten FN, Lund TS. 2006. Kinetic energy conservation issues associated with the collocated mesh scheme for incompressible flow. J. Comput. Phys. 215:465-84 (Pubitemid 43585675)
17. Frank J. 2006. Conservation of wave action under multisymplectic discretizations. J. Phys. A Math. Gen. 39:5479-93 (Pubitemid 43659097)
18. Frank J, Moore BE, Reich S. 2006. Linear PDEs and numerical methods that preserve a multi-symplectic conservation law. SIAM J. Sci. Comput. 28:260-77
19. Frank J, Reich S. 2003. Conservation properties of smoothed particle hydrodynamics applied to the shallow water equations. BIT Numer. Methods 43:40-54
20. Griffiths DF. 1981. An approximately divergence-free 9-node velocity element (with variations) for incompressible flow. Int. J. Num. Methods Fluids 1:323-46
21. Hall CA, Cavendish JC, Frey WH. 1991. The dual variable method for solving fluid flow difference equations on Delaunay triangulations. Comput. Fluids 20(2):145-64 (Pubitemid 21693917)
22. Hall CA, Peterson JS, Porsching TA, Sledge FR. 1985. The dual variable method for finite element discretizations of Navier/Stokes equations. Int. J. Numer. Methods Eng. 21:883-98 (Pubitemid 15514912)
23. Harlow FE, Welch JE. 1965. Numerical calculations of time dependent viscous incompressible flow of fluid with a free surface. Phys. Fluids 8(12):2182-89
24. Harten A. 1983. On the symmetric form of systems of conservation laws with entropy. J. Comput. Phys. 49(1):151-64
25. Hiptmair R. 2001. Discrete Hodge operators: an algebraic perspective. Prog. Electromagn. Res. 32:247-69
26. Honein A, Moin P. 2004. Higher entropy conservation and numerical stability of compressible turbulence simulations. J. Comput. Phys. 201(2):531-45 (Pubitemid 39525418)
27. Hughes TJR, Engel G, Mazzei L, Larson MG. 2000. The continuous Galerkin Method is locally conservative. J. Comput. Phys. 163:467-88
28. Hughes TJR, Franca LP, Mallet M. 1986. A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics. Comput. Methods Appl. Mech. Eng. 54:223-34 (Pubitemid 16504272)
29. Hyman JM, Shashkov M. 1999. The orthogonal decomposition theorems for mimetic finite difference methods. SIAM J. Numer. Anal. 36(3):788-818
30. Ismail F, Roe PL, Nishikawa H. 2006. A proposed cure to the carbuncle phenomenon. In Computational Fluid Dynamics 2006: Proc. 4th Int. Conf. Comput. Fluid Dyn., ICCFD, Ghent, Belgium, 10-14July 2006, ed. H Deconinck, E Dick, pp. 149-54. Berlin: Springer
31. Jameson A. 2008. The construction of discretely conservative finite volume schemes that also globally conserve energy or entropy. J. Sci. Comput. 34(2):152-87
32. Keller HB. 1971. A new difference scheme for parabolic problems. In Numerical Solution of Partial Differential Equations II, ed. B Hubbard, pp. 327-50. New York: Academic
33. Lilly DK. 1965. On the computational stability of numerical solutions of time-dependent nonlinear geophysical fluid dynamics problems. Mon. Weather Rev. 93:11-26
34. Lipnikov K, Shashkov M, Yotov I. 2009. Local flux mimetic finite difference methods. Numer. Math. 112:115-52
35. Mahesh K, Constantinescu G, Moin P. 2004. A numerical method for large-eddy simulation in complex geometries. J. Comput. Phys. 197(1):215-40 (Pubitemid 38739651)
36. Martell M, Perot JB, Rothstein J. 2009. Direct numerical simulation of turbulent flow over drag-reducing ultrahydrophobic surfaces. J. Fluid Mech. 620:31-41
37. Mattiussi C. 1997. An analysis of finite volume, finite element, and finite difference methods using some concepts from algebraic topology. J. Comput. Phys. 133:289-309 (Pubitemid 127168801)
38. Mattiussi C. 2000. The finite volume, finite element and finite difference methods as numerical methods for physical field problems. Adv. Imaging Electron Phys. 113:1-146
39. Margolin LG, Shashkov M. 2007. Finite volume methods and the equations of finite scale: a mimetic approach. Int. J. Numer. Methods Fluids 56(8):991-1002
40. Mavriplis DJ. 1997. Unstructured grid techniques. Annu. Rev. Fluid Mech. 29:473-514
41. McNamara GR, Gareia AL, Aider BJ. 1997. A hydrodynamically correct thermal lattice Boltzmann model. J. Stat. Phys. 87:1111-21 (Pubitemid 127357724)
42. Mittal R, Moin P. 1997. Suitability of upwind-biased finite difference schemes for large-eddy simulation of turbulent flows. AIAA J. 30(8):1415-17 (Pubitemid 127659631)
43. Moin P, Mahesh K. 1998. Direct numerical simulation: a tool in turbulence research. Annu. Rev. Fluid Mech. 30:539-78
44. Morinishi Y. 2010. Skew-symmetric form of convective terms and fully conservative finite difference schemes for variable density low-Mach number flows. J. Comput. Phys. 229(2):276-300
45. Morinishi Y, Lund TS, Vasilyev OV, Moin P. 1998. Fully conservative higher order finite difference schemes for incompressible flow. J. Comput. Phys. 143(1):90-124 (Pubitemid 128348585)
46. 3. Numer. Math. 50:315-41
47. Nicolaides RA. 1993. The covolume approach to computing incompressible flow. In Incompressible Computational Fluid Dynamics, ed. MD Gunzburger, RA Nicolaides, pp. 295-34. Cambridge, UK: Cambridge Univ. Press
48. Oshima M, Hughes TJR, Jansen K. 1998. Consistent finite element calculation of boundary and internal fluxes. Int. J. Comput. Fluid Dyn. 9:227-35 (Pubitemid 128544296)
49. Perot JB. 2000. Conservation properties of unstructured staggered mesh schemes. J. Comput. Phys. 159:58-89
50. Perot JB, Gadebusch J. 2007. A self-adapting turbulence model for flow simulation at any mesh resolution. Phys. Fluids 19(11):115105
51. Perot JB, Nallapati R. 2003. A moving unstructured staggered mesh method for the simulation of incompressible free-surface flows. J. Comput. Phys. 184:192-214 (Pubitemid 36233165)
52. Perot JB, Subramanian V. 2007a. A discrete calculus analysis of the Keller box scheme and a generalization of the method to arbitrary meshes. J. Comput. Phys. 226(1):494-508
53. Perot JB, Subramanian V. 2007b. Discrete calculus methods for diffusion. J. Comput. Phys. 224(1):59-81
54. Perot JB, Vidovic D, Wesseling P. 2006. Mimetic reconstruction of vectors. In Compatible Spatial Discretizations, ed. DN Arnold, PB Bochev, RB Lehoucq, RA Nicolaides, M Shashkov, pp 173-88. IMA Vol. Math. Appl. 142. New York: Springer
55. Perot JB, Zhang X. 1999. Reformulation of the unstructured staggered mesh method as a classic finite volume method. In Finite Volumes for Complex Applications II, ed. R Vilsmeier, F Benkhaldoun, D Hänel, pp. 263-70. Paris: Hermes Sci. Publ.
56. Preissman A. 1961. Propagation des intumescences dans les canaux et rivieres. In 1st Congr. Assoc. Fr. Calc., AFCAL, Grenoble, pp. 433-42
57. Raviart PA, Thomas JM. 1977. A mixed finite element method for second order elliptic problems. In Mathematical Aspects of Finite Element Methods, ed. I Galligani, E Magenes, pp. 292-315. Springer Lect. Notes Math. Vol. 606. New York: Springer-Verlag
58. Rodrigue G, White D. 2001. A vector finite element time-domain method for solving Maxwell equations on unstructured hexahedral grids. SIAM J. Sci. Comput. 23(3):683-706
59. Serrano M, Espanol P. 2001. Thermodynamically consistent mesoscopic fluid particle model. Phys. Rev. E 64:046115
60. Shashkov M, Steinberg S. 1995. Support-operator finite-difference algorithms for general elliptic problems. J. Comput. Phys. 118:131-51
61. Shashkov M, Swartz B, Wendroff B. 1998. Local reconstruction of a vector field from its normal components on the faces of grid cells. J. Comput. Phys. 139:406-9 (Pubitemid 128347112)
62. Strand B. 1994. Summation by parts for finite difference approximations for d/dx. J. Comput. Phys. 110:47-67 (Pubitemid 124013344)
63. Subbareddy PK, Candler GV. 2009. A fully discrete, kinetic energy consistent finite-volume scheme for compressible flows. J. Comput. Phys. 228:1347-64
64. Subramanian V, Perot JB. 2007. Higher-order mimetic methods for unstructured meshes. J. Comput. Phys. 219(1):68-85
65. Sukumar N. 2003. Voronoi cell finite difference method for the diffusion operator on arbitrary unstructured grids. Int. J. Numer. Methods Eng. 57:1-34 (Pubitemid 36624104)
66. Tadmor E. 1984. Skew-self-adjoint form for systems of conservation laws. J. Math. Anal. Appl. 103(2):428-42 (Pubitemid 15457712)
67. Thornber B, Drikakis D, Williams RJR, Youngs D. 2008. On entropy generation and dissipation of kinetic energy in high-resolution shock-capturing schemes. J. Comput. Phys. 227(10):4853-72
68. Tonti E. 1975. On the Formal Structure of Physical Theories. Milano: Consiglio Nazionale delle Ricerche
69. Tonti E. 1976. The reason for analogies between physical theories. Appl. Math. Model. 1:37-50
70. Vasilyev OV. 2000. High order finite difference schemes on nonuniform meshes with good conservation properties. J. Comput. Phys. 157:746-61
71. Vaughan GL, Healy TR, Bryan KR, Sneyd AD, Gorman RM. 2008. Completeness, conservation and error in SPH for fluids. Int. J. Numer. Methods Fluids 56:37-62
72. Verstappen R, Veldman A. 1996. A fourth-order finite volume method for direct numerical simulation of turbulence at higher Reynolds numbers. In Computational Fluid Dynamics '96, ed. J-A Désidéri, C Hirsch, P Le Tallec, M Pandolfi, J Périaux, pp. 1073-79. New York: Wiley & Sons
73. Wenneker I, Segal A, Wesseling P. 2003. Conservation properties of a new unstructured staggered scheme. Comput. Fluids 32:139-47
74. Wesseling P, Segal A, van Kan J, Oosterlee CW, Kassels C. 1992. Finite volume discretization of the incompressible Navier-Stokes equations in general coordinates on staggered grids. Comput. Fluid Dyn. J. 1:27-33
75. White D. 1998. Orthogonal vector basis functions for time domain finite element solution of the vector wave equation. In 8th Bienn. IEEE Conf. Electromagn. Field Comput., Tucson, AZ, UCRL-JC-129188.
76. Livermore, CA: Lawrence Livermore Natl. Lab. Whitney H. 1957. Geometric Integration Theory. Princeton, NJ: Princeton Univ. Press
77. Zhang X, Schmidt D, Perot JB. 2001. Accuracy and conservation properties of a three-dimensional staggered mesh scheme. J. Comput. Phys. 175:764-91