Free Access
Volume 47, Number 5, September-October 2013
Page(s) 1493 - 1513
Published online 01 August 2013
  1. R. Abraham and J.E. Marsden, Foundations of mechanics, 2nd ed. Addison-Wesley (1978). [Google Scholar]
  2. U.M. Ascher, H. Chin and S. Reich, Stabilization of DAEs and invariant manifolds. Numer. Math. 6 (1994) 131–149. [CrossRef] [Google Scholar]
  3. J. Baumgarte, Stabilization of constraints and integrals of motion in dynamical systems. Comput. Math. Appl. Mech. Eng. 1 (1972) 1–16. [Google Scholar]
  4. C.J. Budd, R. Carretero-Gonzalez and R.D. Russell, Precise computations of chemotactic collapse using moving mesh methods. J. Comput. Phys. 202 (2005) 462–487. [Google Scholar]
  5. C.J. Budd and V. Dorodnitsyn, Symmetry adapted moving mesh schemes for the nonlinear Schrodinger equation. J. Phys. A 34 (2001) 10387. [CrossRef] [MathSciNet] [Google Scholar]
  6. C.J. Budd, W.Z. Huang and R.D. Russell, Moving mesh methods for problems with blow-up. SIAM J. Sci. Comput. 17 (1996) 305–327. [CrossRef] [MathSciNet] [Google Scholar]
  7. C.J. Budd, B. Leimkuhler and M.D. Piggott, Scaling invariance and adaptivity. Appl. Numer. Math. 39 (2001) 261–288. [CrossRef] [Google Scholar]
  8. C.J. Budd and M.D. Piggott, Geometric integration and its applications. in Handbook of Numerical Analysis. North-Holland (2000) 35–139. [Google Scholar]
  9. C.J. Budd and J.F. Williams, Parabolic Monge-Ampère methods for blow-up problems in several spatial dimensions. J. Phys. A 39 (2006) 5425–5444. [CrossRef] [MathSciNet] [Google Scholar]
  10. C.J. Budd and J.F. Williams, Moving mesh generation using the parabolic Monge-Ampère equation. SIAM J. Sci. Comput. 31 (2009) 3438–3465. [CrossRef] [Google Scholar]
  11. C.J. Budd and J.F. Williams, How to adaptively resolve evolutionary singularities in differential equations with symmetry. J. Eng. Math. 66 (2010) 217–236. [CrossRef] [Google Scholar]
  12. J.A. Cadzow, Discrete calculus of variations. Internat. J. Control 11 (1970) 393–407. [CrossRef] [Google Scholar]
  13. E. Celledoni, V. Grimm, R.I. McLachlan, D.I. McLaren, D.R.J. O’Neale, B. Owren, and G.R.W. Quispel, Preserving energy resp. dissipation in numerical PDEs, using the average vector field method. NTNU reports, Numerics No 7/09. [Google Scholar]
  14. E. Celledoni, R.I. McLachlan, D.I. McLaren, B. Owren, G.R.W. Quispel and W.M. Wright, Energy-preserving Runge–Kutta methods. ESAIM: M2AN 43 (2009) 645–649. [CrossRef] [EDP Sciences] [Google Scholar]
  15. P. Chartier, E. Faou and A. Murua, An algebraic approach to invariant preserving integrators: The case of quadratic and Hamiltonian invariants. Numer. Math. 103 (2006) 575–590. [CrossRef] [MathSciNet] [Google Scholar]
  16. M. Dahlby and B. Owren, A general framework for deriving integral preserving numerical methods for PDEs. NTNU reports, Numerics No 8/10. [Google Scholar]
  17. M. Dahlby, B. Owren and T. Yaguchi, Preserving multiple first integrals by discrete gradients. J. Phys. A 44 (2011) 305205. [CrossRef] [MathSciNet] [Google Scholar]
  18. V. Dorodnitsyn, Noether-type theorems for difference equations. Appl. Numer. Math. 39 (2001) 307–321. [CrossRef] [Google Scholar]
  19. V. Dorodnitsyn, Applications of Lie Groups to Difference Equations. CRC press, Boca Raton, FL (2010). [Google Scholar]
  20. E. Eich, Convergence results for a coordinate projection method applied to mechanical systems with algebraic constraints. SIAM J. Numer. Anal. 30 (1993) 1467–1482. [CrossRef] [Google Scholar]
  21. K. Feng and M. Qin, Symplectic Geometry Algorithms for Hamiltonian Systems. Springer-Verlag, Berlin (2010). [Google Scholar]
  22. R.C. Fetecau, J.E. Marsden, M. Ortiz and M. West, Nonsmooth Lagrangian mechanics and variational collision integrators. SIAM J. Appl. Dynam. Sys. 2 (2003) 381–416. [CrossRef] [MathSciNet] [Google Scholar]
  23. D. Furihata, Finite difference schemes for equation Formula that inherit energy conservation or dissipation property. J. Comput. Phys. 156 (1999) 181–205. [CrossRef] [Google Scholar]
  24. D. Furihata, A stable and conservative finite difference scheme for the Cahn–Hilliard equation. Numer. Math. 87 (2001) 675–699. [CrossRef] [MathSciNet] [Google Scholar]
  25. D. Furihata, Finite difference schemes for nonlinear wave equation that inherit energy conservation property. J. Comput. Appl. Math. 134 (2001) 35–57. [CrossRef] [Google Scholar]
  26. D. Furihata and T. Matsuo, A Stable, convergent, conservative and linear finite difference scheme for the Cahn–Hilliard equation. Japan J. Indust. Appl. Math. 20 (2003) 65–85. [CrossRef] [MathSciNet] [Google Scholar]
  27. D. Furihata and T. Matsuo, Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations. CRC Press, Boca Raton, FL (2011). [Google Scholar]
  28. H. Goldstein, C. Poole and J. Safko, Classical Mechanics, 3rd ed. Addison-Wesley, New York (2002). [Google Scholar]
  29. O. Gonzalez, Time integration and discrete Hamiltonian systems. J. Nonlinear Sci. 6 (1996) 449–467. [CrossRef] [MathSciNet] [Google Scholar]
  30. E. Hairer, Symmetric projection methods for differential equations on manifolds. BIT 40 (2000) 726–734. [CrossRef] [MathSciNet] [Google Scholar]
  31. E. Hairer, Geometric integration of ordinary differential equations on manifolds. BIT 41 (2001) 996–1007. [CrossRef] [MathSciNet] [Google Scholar]
  32. E. Hairer, Energy-preserving variant of collocation methods. J. Numer. Anal. Ind. Appl. Math. 5 (2010) 73–84. [Google Scholar]
  33. E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd ed. Springer-Verlag, Berlin (2006). [Google Scholar]
  34. W. Huang, Y. Ren and R.D. Russell, Moving mesh partial differential equations (MMPDES) based on the equidistribution principle. SIAM J. Numer. Anal. 31 (1994) 709–730. [CrossRef] [Google Scholar]
  35. P.E. Hydon and E.L. Mansfield, A variational complex for difference equations. Found. Comput. Math. 4 (2004) 187–217. [CrossRef] [MathSciNet] [Google Scholar]
  36. T. Itoh and K. Abe, Hamiltonian-conserving discrete canonical equations based on variational difference quotients. J. Comput. Phys. 76 (1988) 85–102. [CrossRef] [Google Scholar]
  37. C. Kane, J.E. Marsden, M. Ortiz and M. West, Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems. Int. J. Numer. Methods Eng. 49 (2000) 1295–1325. [CrossRef] [Google Scholar]
  38. C.T. Kelley, Solving nonlinear equations with Newton’s method. SIAM, Philadelphia (2003). [Google Scholar]
  39. R.A. LaBudde and D. Greenspan, Discrete mechanics—a general treatment. J. Comput. Phys. 15 (1974) 134–167. [CrossRef] [Google Scholar]
  40. R.A. LaBudde and D. Greenspan, Energy and momentum conserving methods of arbitrary order of the numerical integration of equations of motion I. Motion of a single particle. Numer. Math. 25 (1976) 323–346. [CrossRef] [Google Scholar]
  41. R.A. LaBudde and D. Greenspan, Energy and momentum conserving methods of arbitrary order for the numerical integration of equations of motion II. Motion of a system of particles. Numer. Math. 26 (1976) 1–16. [CrossRef] [MathSciNet] [Google Scholar]
  42. L.D. Landau and E.M. Lifshitz, Mechanics, 3rd ed. Butterworth-Heinemann, London (1976). [Google Scholar]
  43. R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002). [Google Scholar]
  44. A. Lew, J.E. Marsden, M. Ortiz and M. West, Asynchronous variational integrators. Arch. Ration. Mech. Anal. 167 (2003) 85–146. [CrossRef] [MathSciNet] [Google Scholar]
  45. S. Li and L. Vu-Quoc, Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein–Gordon equation. SIAM J. Numer. Anal. 32 (1995) 1839–1875. [CrossRef] [Google Scholar]
  46. J.D. Logan, First integrals in the discrete variational calculus. Aequationes Math. 9 (1973) 210–220. [CrossRef] [MathSciNet] [Google Scholar]
  47. E.L. Mansfield and G.R.W. Quispel, Towards a variational complex for the finite element method. Group theory and numerical analysis. In CRM Proc. of Lect. Notes Amer. Math. Soc. Providence, RI 39 (2005) 207–232. [Google Scholar]
  48. J.E. Marsden, G.W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs. Commun. Math. Phys. 199 (1998) 351-395. [Google Scholar]
  49. J.E. Marsden, S. Pekarsky, S. Shkoller and M. West, Variational methods, multisymplectic geometry and continuum mechanics. J. Geom. Phys. 38 (2001) 253–284. [CrossRef] [MathSciNet] [Google Scholar]
  50. J.E. Marsden and M. West, Discrete mechanics and variational integrators. Acta Numer. 10 (2001) 357–514. [CrossRef] [MathSciNet] [Google Scholar]
  51. T. Matsuo, High-order schemes for conservative or dissipative systems. J. Comput. Appl. Math. 152 (2003) 305–317. [CrossRef] [Google Scholar]
  52. T. Matsuo, New conservative schemes with discrete variational derivatives for nonlinear wave equations. J. Comput. Appl. Math. 203 (2007) 32–56. [CrossRef] [Google Scholar]
  53. T. Matsuo, Dissipative/conservative Galerkin method using discrete partial derivative for nonlinear evolution equations. J. Comput. Appl. Math. 218 (2008) 506–521. [CrossRef] [Google Scholar]
  54. T. Matsuo and D. Furihata, Dissipative or conservative finite difference schemes for complex-valued nonlinear partial differential equations. J. Comput. Phys. 171 (2001) 425–447. [CrossRef] [MathSciNet] [Google Scholar]
  55. T. Matsuo, M. Sugihara, D. Furihata and M. Mori, Linearly implicit finite difference schemes derived by the discrete variational method. RIMS Kokyuroku 1145 (2000) 121–129. [Google Scholar]
  56. T. Matsuo, M. Sugihara, D. Furihata and M. Mori, Spatially accurate dissipative or conservative finite difference schemes derived by the discrete variational method. Japan J. Indust. Appl. Math. 19 (2002) 311–330. [CrossRef] [MathSciNet] [Google Scholar]
  57. R.I. McLachlan, G.R.W. Quispel and N. Robidoux, Geometric integration using discrete gradients. Philos. Trans. Roy. Soc. A 357 (1999) 1021–1046. [CrossRef] [Google Scholar]
  58. R.I. McLachlan and N. Robidoux, Antisymmetry, pseudospectral methods, weighted residual discretizations, and energy conserving partial differential equations, preprint. [Google Scholar]
  59. K.S. Miller, Linear difference equations, W.A. Benjamin Inc., New York–Amsterdam (1968). [Google Scholar]
  60. P. Olver, Applications of Lie Groups to Differential Equations, 2nd ed. In vol. 107. Graduate Texts in Mathematics. Springer-Verlag, New York (1993). [Google Scholar]
  61. F.A. Potra and W.C. Rheinboldt, On the numerical solution of Euler − Lagrange equations. Mech. Struct. Mach., 19 (1991) 1–18. [CrossRef] [Google Scholar]
  62. F.A. Potra and J. Yen, Implicit numerical integration for Euler − Lagrange equations via tangent space parametrization. Mech. Struct. Mach. 19 (1991) 77–98. [CrossRef] [Google Scholar]
  63. G.R.W. Quispel and D.I. McLaren, A new class of energy-preserving numerical integration methods. J. Phys. A 41 (2008) 045206. [CrossRef] [MathSciNet] [Google Scholar]
  64. I. Saitoh, Symplectic finite difference time domain methods for Maxwell equations -formulation and their properties-. In Book of Abstracts of SciCADE 2009 (2009) 183. [Google Scholar]
  65. J.M. Sanz-Serna and M.P. Calvo, Numerical Hamiltonian Problems. In vol. 7 of Applied Mathematics and Mathematical Computation. Chapman and Hall, London (1994). [Google Scholar]
  66. L.F. Shampine, Conservation laws and the numerical solution of ODEs. Comput. Math. Appl. B12 (1986) 1287–1296. [CrossRef] [Google Scholar]
  67. L.F. Shampine, Conservation laws and the numerical solution of ODEs II. Comput. Math. Appl. 38 (1999) 61–72. [CrossRef] [Google Scholar]
  68. M. West, Variational integrators, Ph.D. thesis, California Institute of Technology (2004). [Google Scholar]
  69. M. West, C. Kane, J.E. Marsden and M. Ortiz, Variational integrators, the Newmark scheme, and dissipative systems. In EQUADIFF 99 (Vol. 2): Proc. of the International Conference on Differential Equations. World Scientific (2000) 1009–1011. [Google Scholar]
  70. T. Yaguchi, T. Matsuo and M. Sugihara, An extension of the discrete variational method to nonuniform grids. J. Comput. Phys. 229 (2010) 4382–4423. [CrossRef] [Google Scholar]
  71. G. Zhong and J.E. Marsden, Lie–Poisson integrators and Lie–Poisson Hamilton–Jacobi theory. Phys. Lett. A 133 (1988) 134–139. [CrossRef] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you