A fourth-order auxiliary variable projection method for zero-Mach number gas dynamics

Journal of Computational Physics

Volumn 227, Issue 3, 2008, Pages 2012-2043

  Kadioglu, Samet Y ^a   Klein, Rupert ^b   Minion, Michael L ^c  

^a IDAHO NATIONAL LABORATORY   (United States)

^b FREIE UNIVERSITÄT BERLIN   (Germany)

^c University of North Carolina at Chapel Hill   (United States)

Author keywords

Auxiliary variable methods; Deferred corrections; Gas dynamics; Gauge methods; Impulse methods; Projection methods

Indexed keywords

AERODYNAMICS; COMPUTATIONAL FLUID DYNAMICS; FINITE VOLUME METHOD; GAGES; INCOMPRESSIBLE FLOW; NUMERICAL METHODS;

AUXILIARY VARIABLE METHOD; AUXILIARY VARIABLES; CONSERVATION EQUATIONS; DEFERRED CORRECTION; FOURTH-ORDER; GAUGE METHOD; IMPULSE METHODS; PROJECTION METHOD; VARIABLE FORMULATION; VARIABLE PROJECTION METHODS;

MACH NUMBER;

EID: 37249061618     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2007.10.008     Document Type: Article

References (78)

1. Almgren A.S., Bell J.B., Colella P., Howell L.H., and Welcome M.L. A conservative adaptive projection method for the variable density incompressible Navier-Stokes equations. J. Comput. Phys. 142 (1998) 1-46
2. Almgren A.S., Bell J.B., Rendleman C.A., and Zingale M. Low mach number modeling of type Ia supernovae. I. Hydrodynamics. Astrophys. J. 637 (2006) 922-936
3. Almgren A.S., Bell J.B., Rendleman C.A., and Zingale M. Low mach number modeling of type Ia supernovae. II. Energy evolution. Astrophys. J. 649 (2006) 927-938
4. Bell J.B., Colella P., and Glaz H.M. A second order projection method for the incompressible Navier-Stokes equations. J. Comput. Phys. 85 2 (1989) 257-283
5. J.B. Bell, M.S. Day, A.S. Almgren, M.J. Lijewski, C.A. Rendleman, Adaptive numerical simulation of turbulent premixed combustion, in: Proceedings of the First MIT Conference on Computational Fluid and Solid Mechanics, June 2001.
6. Bell J.B., Day M.S., Shepherd I.G., Johnson M., Cheng R.K., Grcar J.F., Beckner V.E., and Lijewski M.J. Numerical simulation of a laboratory-scale turbulent V-flame. Proc. Natl. Acad. Sci. USA 102 (2005) 10006-10011
7. Bell J.B., and Marcus D.L. A second order projection method for variable density flows. J. Comput. Phys. 101 1 (1992) 1-24
8. Bijl H., and Wesseling P. A unified method for computing incompressible and compressible flows in boundary-fitted coordinates. J. Comput. Phys. 141 (1998) 153-173
9. Botta N., Klein R., and Almgren A.S. Asymptotic analysis of a dry atmosphere. In: Neittaanmäki P., Tiihonen T., and Tarvainen P. (Eds). ENUMATH 99 - Proceedings of the 3rd European Conference on Numerical Mathematics, Jyväskylä, Finland, 1999 (2000), World Scientific, Singapore
10. Bourlioux A., Layton A.T., and Minion M.L. Higher-order multi-implicit spectral deferred correction methods for problems of reacting flow. J. Comput. Phys. 189 (2003) 351-376
11. Brown D.L., Cortez R., and Minion M.L. Accurate projection methods for the incompressible Navier-Stokes equations. J. Comput. Phys. 168 (2001) 464-499
12. Buttke T. Velicity methods: Lagrangian numerical methods which preserve the Hamiltonian structure of incompressible fluid flow. NATO ASI Series C vol. 395 (1993), Kluwer Academic Publisher 39-57
13. Buttke T., and Chorin A.J. Turbulence calculations in magnetization variables. Appl. Numer. Math. 12 (1993) 47-54
14. Chorin A.J. A numerical method for solving incompressible viscous flow problems. J. Comput. Phys. 2 (1967) 12-26
15. Chorin A.J. Numerical solution of the Navier-Stokes equations for an incompressible fluid. Bull. Am. Math. Soc. 73 (1967) 928-931
16. Chorin A.J. Numerical solution of incompressible flow problems. Stud. Numer. Anal. 2 (1968) 64-71
17. Chorin A.J. Numerical solution of the Navier-Stokes equations. Math. Comput. 22 (1968) 742-762
18. Colella P. Multidimensional upwind methods for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 171-200
19. Colella P., and Woodward P.R. The piecewise parabolic method (ppm) for gas-dynamics simulations. J. Comput. Phys. 54 (1984) 174-201
20. Ricardo Cortez, Impulse-based Particle Methods for Fluid Flow, Ph.D. Thesis, University of California, Berkeley, May 1995.
21. Cortez R. An impulse-based approximation of fluid motion due to boundary forces. J. Comput. Phys. 123 (1996) 341-353
22. Cortez R. On the accuracy of impulse methods for fluid flow. SIAM J. Sci. Comput. 19 4 (1998) 1290-1302
23. Durran D.R. Numerical Methods for Wave Equations in Geophysical Fluid Dynamics (1999), Springer Verlag, New York, Berlin, Heidelberg
24. Dutt A., Greengard L., and Rokhlin V. Spectral deferred correction methods for ordinary differential equations. BIT 40 2 (2000) 241-266
25. Wienan E., and Liu J.-G. Finite difference schemes for incompressible flows in the velocity-impulse density formulation. J. Comput. Phys. 130 (1997) 67-76
26. W. E, Jian-Guo Liu, Gauge method for viscous incompressible flows, Commun. Math. Sci. 1 (2003) 317-332.
27. K.J. Geratz, Erweiterung eines Godunov Typ Verfahrens für zweidimensionale kompressible Strömungen auf die Fälle kleiner und verschwindender Machzahl, Ph.D. Thesis, Rheinisch Westfälische Technische Hochschule, Aachen, 1997.
28. Gresho P.M. On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly-consistent mass matrix. Int. J. Numer. Methods Fluids 11 (1990) 587
29. Gresho P.M., and Chan S.T. On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly-consistent mass matrix: part 2: Implementation. Int. J. Numer. Methods Fluids 11 (1990) 621-659
30. Guermond J.L., Minev P., and Shen J. An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng. 195 (2006)
31. Gustafsson B., and Kress W. Deferred correction methods for initial value problems. BIT 41 (2001) 986-995
32. Karki K.C., and Patankar S.V. Pressure based calculation procedure for viscous flows at all speeds in arbitrary configurations. AIAA J. 27 (1989) 1167-1174
33. Kim J., and Moin P. Application of a fractional-step method to incompressible Navier-Stokes equations. J. Comput. Phys. 59 (1985) 308-323
34. Klein R. Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics I: One-dimensional flow. J. Comput. Phys. 121 (1995) 213-237
35. Klein R. Asymptotic analyses for atmospheric flows and the construction of asymptotically adaptive numerical methods. ZAMM 80 (2000) 765-777
36. Knio O.M., Najm H.N., and Wyckoff P.S. A semi-implicit numerical scheme for reacting flow. ii stiff, operator-split formulation. JCP 154 (1999) 428-467
37. O. Knoth, Compressible atmospheric modelling at all scales, in: Mathematical Theory and Modelling in Atmosphere-Ocean Science, 2006.
38. Kress W., and Gustafsson B. Deferred correction methods for initial boundary value problems. J. Sci. Comput. 17 1-4 (2002) 241-251
39. Lang J. Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems. Theory, Algorithm, and Applications (2000), Springer-Verlag
40. Layton A.T., and Minion M.L. Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics. J. Comput. Phys. 194 2 (2004) 697-714
41. Layton A.T., and Minion M.L. Implications of the choice of quadrature nodes for Picard integral deferred corrections methods for ordinary differential equations. BIT 45 (2005) 341-373
42. Layton A.T., and Minion M.L. Implications of the choice of predictors for semi-implicit Picard integral deferred corrections methods. Commun. Appl. Math. Comput. Sci. 2 1 (2007) 1-34
43. Lipps F., and Hemler R. A scale analysis of deep moist convection and some related numerical calculations. JAS 29 (1982) 2192-2210
44. Liska R., and Wendroff B. Comparison of several finite difference schemes on 1D and 2D test problems for the Euler equations. SIAM J. Sci. Comput. 25 3 (2003) 995-1017
45. Liu M., Ren Y., and Zhang H. A class of fully second order accurate projection methods for solving the incompressible Navier-Stokes equations. J. Comput. Phys. 200 1 (2004) 325-346
46. Maddocks J.H., and Pego R.L. An unconstrained hamiltonian formulation for incompressible fluid flow. Commun. Math. Phys. 170 (1995) 207-217
47. Majda A., and Sethian J. The derivation and numerical solution of the equations for zero Mach number combustion. Combust. Sci. Technol. 42 (1985) 185-205
48. Minion M.L. Semi-implicit projection methods for incompressible flow based on spectral deferred corrections. Appl. Numer. Math. 48 3-4 (2004) 369-387
49. Minion M.L. A note on the stability of Godunov-projection methods. J. Comput. Phys. 123 (1996)
50. Michael L. Minion, Higher-order semi-implicit projection methods, in: M. Hafez, (Ed.), Numerical Simulations of Incompressible Flows. Papers from the workshop held in Half Moon Bay, CA, June 19-21, 2001,World Scientific Publishing, River Edge, NJ, January 2003, pp. 126-140.
51. Minion M.L. Semi-implicit spectral deferred correction methods for ordinary differential equations. Commun. Math. Sci. 1 (2003) 471-500
52. Munz C.D., Roller S., Klein R., and Geratz K.J. The extension of incompressible flow solvers to the weakly compressible regime. CandF 32 (2003) 173-196
53. Najm H.N. A conservative low Mach number projection method for reacting flow modeling. In: Chan S.H. (Ed). Transport Phenomena in Combustion vol. 2 (1996), Taylor and Francis, Washington, DC 921-932
54. Najm H.N., Wyckoff P.S., and Knio O.M. A semi-implicit numerical scheme for reacting flow: I stiff chemistry. JCP 143 (1998) 381-402
55. Nerinckx K., Vierendeels J., and Dick E. Mach-uniformity through the coupled pressure and temperature correction algorithm. J. Comput. Phys. 206 (2005) 597-623
56. Ogura Y., and Phillips N. Scale analysis for deep and shallow convection in the atmosphere. JAS 19 (1962) 173-179
57. Oseledets V.I. On a new way of writing the Navier-Stokes equation: The Hamiltonian formalism. Commun. Moscow Math. Soc. 44 1 (1989) 210-211
58. Park J.H., and Munz C.D. Multiple pressure variables methods for fluid flow at all Mach numbers. Int. J. Numer. Methods Fluids 49 (2005) 905-931
59. Patankar S.V., and Spalding D.B. A calculation procedure for heat, mass, and momentum transfer in three-dimensional parabolic flow. Int. J. Heat Mass Transfer 15 (1972) 1787-1806
60. Pereyra V. Iterated deferred corrections for nonlinear operator equations. Numer. Math. 10 (1966) 316-323
61. Pereyra V. On improving an approximate solution of a functional equation by deferred corrections. Numer. Math. 8 (1966) 376-391
62. Puckett E.G., Almgren A.S., Bell J.B., Marcus D.L., and Rider W.J. A second-order projection method for tracking fluid interfaces in variable density incompressible flows. J. Comput. Phys. 130 (1997) 269-282
63. Recchioni M.C., and Russo G. Hamiltonian-based numerical methods for a fluid-membrane interaction in two and three dimensions. SIAM J. Sci. Comput. 19 (1998) 861-892
64. Reisner J.M., Knoll D.A., and Wyszogradzki A.A. An implicitly balanced hurricane model with physics based preconditioning. Monthly Weather Rev. 133 (2005) 1003-1022
65. Ren Y., Jiang Y., Liu M., and Zhang H. A class of fully third-order accurate projection methods for solving the incompressible Navier-Stokes equations. Acta Mech. Sin. 21 6 (2006) 542-549
66. Russo G., and Smereka P. Impulse formulation of the Euler equations: general properties and numerical methods. J. Fluid Mech. 391 (1999) 189-209
67. Th. Schneider, N. Botta, K.J. Geratz, R. Klein, Extension of finite volume compressible flow solvers to multi-dimensional, variable density zero Mach number flow. Technical Report SC 98-31, ZIB, Konrad-Zuse-Zentrum für Informationstechnik Berlin, 1998.
68. Schneider Th., Botta N., Geratz K.J., and Klein R. Extension of finite volume compressible flow solvers to multi-dimensional, variable density zero Mach number flow. J. Comput. Phys. 155 (1999) 248-286
69. Jiang G.S., and Shu C.-W. Efficient implementation of weighted eno schemes. J. Comput. Phys. 126 (1996) 202-228
70. C. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws.
71. Summers D.M., and Chorin A.J. Numerical vorticity creation based on impulse conservation. Proc. Natl. Acad. Sci. USA 93 (1996) 1881-1885
72. Temam R. Sur lapproximation de la solution des equations de Navier-Stokes par le methode des fractionaire II. Arch. Rational Mech. Anal. 33 (1969) 377-385
73. van der Heul D.R., Vuik C., and Wesseling P. Conservative pressure-correction method for flow at all speeds. Comput. Fluids 32 (2003) 1113-1132
74. Van Kan J. A second-order accurate pressure-correction scheme for viscous incompressible flow. SIAM J. Sci. Comput. 7 3 (1986) 870-891
75. Van Leer B. Towards the ultimate conservative difference scheme IV. A new approach to numerical convection. J. Comput. Phys. 23 (1977) 276-299
76. S. Vater, R. Klein, Stability of a cartesian grid projection method for zero froude number shallow water flows, Numerische Mathematik (2007), submitted for publication.
77. Wesseling P. Principles of Computational Fluid Dynamics (2001), Springer-Verlag, Berlin, Heidelberg, New York
78. P.E. Zadunaisky, A method for the estimation of errors propagated in the numerical solution of a system of ordinary differential equations, in: G. Contopoulus (Ed.), The Theory of Orbits in the Solar System and in Stellar Systems, Proceedings of International Astronomical Union, Symposium, vol. 25, 1964.