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Discrete gradient methods have an energy conservation law
1. | Institute of Fundamental Sciences, Massey University, Private Bag 11-222, Palmerston North, New Zealand |
2. | Department of Mathematics and Statistics, La Trobe University, Melbourne, Victoria 3086, Australia |
References:
[1] |
T. J. Bridges and S. Reich, Numerical methods for Hamiltonian PDEs,, J. Phys. A, 39 (2006), 5287.
doi: 10.1088/0305-4470/39/19/S02. |
[2] |
E. Celledoni, V. Grimm, R. I. McLachlan, D. I. McLaren, D. O'Neale, B. Owren and G. R. W. Quispel, Preserving energy resp. dissipation in numerical PDEs using the "Average Vector Field'' method,, J. Comput. Phys., 231 (2012), 6770.
doi: 10.1016/j.jcp.2012.06.022. |
[3] |
O. Gonzalez, Time integration and discrete Hamiltonian systems,, J. Nonlinear Sci., 6 (1996), 449.
doi: 10.1007/BF02440162. |
[4] |
P. E. Hydon and E. L. Mansfield, A variational complex for difference equations,, Found. Comput. Math., 4 (2004), 187.
doi: 10.1007/s10208-002-0071-9. |
[5] |
R. I. McLachlan, G. R. W. Quispel and N. Robidoux, Geometric integration using discrete gradients,, Phil. Trans. Roy. Soc. A, 357 (1999), 1021.
doi: 10.1098/rsta.1999.0363. |
[6] |
M. Oliver and C. Wulff, A-stable Runge-Kutta methods for semilinear evolution equations,, J. Funct. Anal., 263 (2012), 1981.
doi: 10.1016/j.jfa.2012.06.022. |
[7] |
G. R. W. Quispel and D. I. McLaren, A new class of energy-preserving numerical integration methods,, J. Phys. A, 41 (2008).
doi: 10.1088/1751-8113/41/4/045206. |
[8] |
G. R. W. Quispel and G. S. Turner, Discrete gradient methods for solving ODE's numerically while preserving a first integral,, J. Phys. A, 29 (1996).
doi: 10.1088/0305-4470/29/13/006. |
[9] |
E. Rothe, Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben,, Math. Ann., 102 (1930), 650.
doi: 10.1007/BF01782368. |
[10] |
B. N. Ryland, R. I. McLachlan and J. Frank, On multisymplecticity of partitioned Runge-Kutta and splitting methods,, Int. J. Comput. Math., 84 (2007), 847.
doi: 10.1080/00207160701458633. |
[11] |
J. C. Simo and N. Tarnow, The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics,, ZAMP, 43 (1992), 757.
doi: 10.1007/BF00913408. |
[12] |
J. Vanneste, On the derivation of fluxes for conservation laws in Hamiltonian systems,, IMA J. Appl. Math., 59 (1997), 211.
doi: 10.1093/imamat/59.2.211. |
show all references
References:
[1] |
T. J. Bridges and S. Reich, Numerical methods for Hamiltonian PDEs,, J. Phys. A, 39 (2006), 5287.
doi: 10.1088/0305-4470/39/19/S02. |
[2] |
E. Celledoni, V. Grimm, R. I. McLachlan, D. I. McLaren, D. O'Neale, B. Owren and G. R. W. Quispel, Preserving energy resp. dissipation in numerical PDEs using the "Average Vector Field'' method,, J. Comput. Phys., 231 (2012), 6770.
doi: 10.1016/j.jcp.2012.06.022. |
[3] |
O. Gonzalez, Time integration and discrete Hamiltonian systems,, J. Nonlinear Sci., 6 (1996), 449.
doi: 10.1007/BF02440162. |
[4] |
P. E. Hydon and E. L. Mansfield, A variational complex for difference equations,, Found. Comput. Math., 4 (2004), 187.
doi: 10.1007/s10208-002-0071-9. |
[5] |
R. I. McLachlan, G. R. W. Quispel and N. Robidoux, Geometric integration using discrete gradients,, Phil. Trans. Roy. Soc. A, 357 (1999), 1021.
doi: 10.1098/rsta.1999.0363. |
[6] |
M. Oliver and C. Wulff, A-stable Runge-Kutta methods for semilinear evolution equations,, J. Funct. Anal., 263 (2012), 1981.
doi: 10.1016/j.jfa.2012.06.022. |
[7] |
G. R. W. Quispel and D. I. McLaren, A new class of energy-preserving numerical integration methods,, J. Phys. A, 41 (2008).
doi: 10.1088/1751-8113/41/4/045206. |
[8] |
G. R. W. Quispel and G. S. Turner, Discrete gradient methods for solving ODE's numerically while preserving a first integral,, J. Phys. A, 29 (1996).
doi: 10.1088/0305-4470/29/13/006. |
[9] |
E. Rothe, Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben,, Math. Ann., 102 (1930), 650.
doi: 10.1007/BF01782368. |
[10] |
B. N. Ryland, R. I. McLachlan and J. Frank, On multisymplecticity of partitioned Runge-Kutta and splitting methods,, Int. J. Comput. Math., 84 (2007), 847.
doi: 10.1080/00207160701458633. |
[11] |
J. C. Simo and N. Tarnow, The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics,, ZAMP, 43 (1992), 757.
doi: 10.1007/BF00913408. |
[12] |
J. Vanneste, On the derivation of fluxes for conservation laws in Hamiltonian systems,, IMA J. Appl. Math., 59 (1997), 211.
doi: 10.1093/imamat/59.2.211. |
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