Geometric, variational discretization of continuum theories

Physica D: Nonlinear Phenomena

Volumn 240, Issue 21, 2011, Pages 1724-1760

  Gawlik, E S ^a   Mullen, P ^b   Pavlov, D ^c   Marsden, J E ^b   Desbrun, M ^b  

^a STANFORD UNIVERSITY   (United States)

^b California Institute of Technology   (United States)

^c EPFL   (Switzerland)

Author keywords

Complex fluids; Fluid dynamics; Geometric discretization; Magnetohydrodynamics; Structure preserving schemes

Indexed keywords

ALGEBRA; CONTINUUM MECHANICS; FLOW OF FLUIDS; FLUID DYNAMICS; GEOMETRY; LIQUIDS; NONLINEAR EQUATIONS; NUMERICAL METHODS;

COMPLEX FLUIDS; DISCRETIZATIONS; FINITE DIMENSIONAL APPROXIMATION; NUMERICAL IMPLEMENTATION; SPATIAL AND TEMPORAL RESOLUTIONS; STRUCTURE-PRESERVING; TEMPORAL DISCRETIZATION; VARIATIONAL DISCRETIZATION;

MAGNETOHYDRODYNAMICS;

EID: 80052936964     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2011.07.011     Document Type: Article

References (47)

1. E. Hairer, C. Lubich, and G. Wanner Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations 2006 Springer Berlin
2. J.E. Marsden, and M. West Discrete mechanics and variational integrators Acta Numer. 10 2001 357 514 (Pubitemid 33305814)
3. J.U. Brackbill, and D.C. Barnes The effect of nonzero product of magnetic gradient and B on the numerical solution of the magnetohydrodynamic equations J. Comput. Phys. 35 1980 426 430
4. A.L. Zachary, A. Malagoli, and P. Colella A higher-order Godunov method for multidimensional ideal magnetohydrodynamics SIAM J. Sci. Comput. 15 2 1994 263 284
5. J.G. Liu, and W.C. Wang An energy-preserving MAC-Yee scheme for the incompressible MHD equation J. Comput. Phys. 174 2001 12 37
6. D.A. Clarke, M.L. Norman, and J.O. Burns Numerical simulations of a magnetically confined jet Astrophys. J. 311 1986 L63 L67
7. K.G. Powell, An approximate Riemann solver for magnetohydrodynamics (that works in more than one dimension), ICASE Report No. 94-24, 1994.
8. A. Lew, J.E. Marsden, M. Ortiz, and M. West An overview of variational integrators Finite Element Methods: 1970's and Beyond 2004 98 115
9. P.E. Hydon Multisymplectic conservation laws for differential and differential-difference equations Proc. R. Soc. London, Ser. A 461 2005 1627 1637 (Pubitemid 40991894)
10. J.E. Marsden, G.W. Patrick, and S. Shkoller Multisymplectic geometry, variational integrators, and nonlinear PDEs Comm. Math. Phys. 199 1998 351 355
11. R. Hiptmair Finite elements in computational electromagnetism Acta Numer. 11 2002 237 339
12. P.B. Bochev, and J.M. Hyman Principles of mimetic discretizations of differential operators IMA 142 2006 89 119
13. D.N. Arnold, R.S. Falk, and R. Winther Finite element exterior calculus, homological techniques, and applications Acta Numer. 15 2006 1 155 (Pubitemid 43748008)
14. M. Desbrun, E. Grinspun, P. Schröder, and M. Wardetzky Discrete Differential Geometry: An Applied Introduction SIGGRAPH Asia 2008 Course Notes 2008 46 62 (Chapter 7)
15. J.P. Goedbloed, and S. Poedts Principles of Magnetohydrodynamics 2004 Cambridge University Press Cambridge
16. B. Perot Conservation properties of unstructured staggered mesh schemes J. Comput. Phys. 159 2000 58 89
17. X. Zhang, D.B. Schmidt, and B. Perot Accuracy and conservation properties of a 3D unstructured staggered mesh scheme for fluid dynamics J. Comput. Phys. 175 2002 764 791
18. P. Mullen, K. Crane, D. Pavlov, Y. Tong, and M. Desbrun Energy-preserving integrators for fluid animation ACM Trans. Graph. 28 3 2009
19. S. Elcott, Y. Tong, E. Kanso, P. Schröder, and M. Desbrun Stable, circulation-preserving, simplicial fluids ACM Trans. Graph. 26 2007 1 18
20. C.J. Cotter, D.D. Holm, and P.E. Hydon Multisymplectic formulation of fluid dynamics using the inverse map Proc. R. Soc. Lond. Ser. A 463 2007 2671 2687
21. D. Pavlov, P. Mullen, Y. Tong, E. Kanso, J.E. Marsden, and M. Desbrun Structure-preserving discretization of incompressible fluids Physica D 240 6 2011 443 458
22. A. Stern, Y. Tong, M. Desbrun, and J.E. Marsden Variational integrators for Maxwell's equations with sources PIERS Online 4 2008 711 715
23. K.S. Yee Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media IEEE Trans. Antennas Propag. 14 3 1966 302 307
24. A. Bossavit, and L. Kettunen Yee-like schemes on a tetrahedral mesh with diagonal lumping Int. J. Numer. Anal. Model. 12 12 1999 129 142
25. J. Moser, and A.P. Veselov Discrete versions of some classical integrable systems and factorization of matrix polynomials Int. J. Numer. Anal. Model. 139 1999 217 243
26. A.I. Bobenko, and Y.B. Suris Discrete time Lagrangian mechanics on Lie groups, with an application to the Lagrangec top Comm. Math. Phys. 204 1999 147 188
27. N. Bou-Rabee, and J.E. Marsden HamiltonPontryagin integrators on Lie groups part I: introduction and structure-preserving properties Found. Comput. Math. 9 2009 197 219
28. V.I. Arnold Sur la géometrie differentielle des groupes de Lie de dimension infinie et ses applications l'hydrodynamique des fluides parfaits Ann. Inst. Fourier 16 1966 319 361
29. J.E. Marsden, D.G. Ebin, A.E. Fischer, Diffeomorphism groups, hydrodynamics, and relativity, in: J. Vanstone (Ed.), Proceedings of the 13th Biennial Seminar of Canadian Mathematical Congress, 1972, pp. 135279.
30. B.O. Koopman Hamiltonian systems and transformation in Hilbert space Proc. Natl. Acad. Sci. 17 1931 315 318
31. V.I. Arnold, and B.A. Khesin Topological Methods in Hydrodynamics 1998 Springer New York
32. D.D. Holm, J.E. Marsden, and T.S. Ratiu The EulerPoincaré equations and semidirect products with applications to continuum theories Adv. Math. 137 1998 1 81 (Pubitemid 128347585)
33. A.M. Bloch Nonholonomic Mechanics and Control 2003 Springer-Verlag New York
34. N. Bou-Rabee, HamiltonPontryagin integrators on Lie groups, Ph.D. Thesis, California Institute of Technology, Pasadena, CA, 2007.
35. J.E. Marsden, T.S. Ratiu, and A. Weinstein Semidirect products and reduction in mechanics Trans. Amer. Math. Soc. 281 1984 147 177
36. D. Holm The EulerPoincaré framework for modeling fluid dynamics Geometric Mechanics and Symmetry: The Peyresq Lectures London Mathematical Society Lecture Note Series 2005 Cambridge University Press 157 209 (Chapter III)
37. F. Gay-Balmaz, and T.S. Ratiu The geometric structure of complex fluids Adv. in Appl. Math. 42 2009 176 275
38. F. Gay-Balmaz, and C. Tronci Reduction theory for symmetry breaking with applications to nematic systems Physica D 239 2010 1929 1947
39. M. Desbrun, A.N. Hirani, M. Leok, J.E. Marsden, Discrete exterior calculus, Preprint, 2005, pp. 153.
40. M. Kobilarov, K. Crane, and M. Desbrun Lie group integrators for animation and control of vehicles ACM Trans. Graph. 28 2009 1 11
41. H.C. Yee, N.D. Sandham, and M.J. Djomehri Low-dissipative high-order shock-capturing methods using characteristic-based filters J. Comput. Phys. 150 1999 199 238
42. T.A. Gardiner, and J.M. Stone An unsplit Godunov method for ideal MHD via constrained transport J. Comput. Phys. 205 2005 509 539 (Pubitemid 40518385)
43. D.S. Balsara, and D.S. Spicer A staggered mesh algorithm using high order Godunov fluxes to ensure solenodial magnetic fields in magnetohydrodynamic simulations J. Comput. Phys. 149 1999 270 292
44. D. Cordoba, and C. Marliani Evolution of current sheets and regularity of ideal incompressible magnetic fluids in 2D Comm. Pure Appl. Math. 53 2000 512 524
45. P. Mullen, A. McKenzie, D. Pavlov, L. Durant, Y. Tong, E. Kanso, J.E. Marsden, and M. Desbrun Discrete Lie advection of differential forms Found. Comput. Math. 11 2 2011 131 149
46. R.J. LeVeque Finite Volume Methods for Hyperbolic Problems 2002 Cambridge University Press
47. R. Abraham, J.E. Marsden, and T.S. Ratiu Manifolds, Tensor Analysis, and Applications 1988 Springer New York