American Institute of Mathematical Sciences

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March  2014, 34(3): 1099-1104. doi: 10.3934/dcds.2014.34.1099

Discrete gradient methods have an energy conservation law

Robert I. McLachlan 1, and  G. R. W. Quispel 2,

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

Received  January 2013 Revised  April 2013 Published  August 2013

We show for a variety of classes of conservative PDEs that discrete gradient methods designed to have a conserved quantity (here called energy) also have a time-discrete conservation law. The discrete conservation law has the same conserved density as the continuous conservation law, while its flux is found by replacing all derivatives of the conserved density appearing in the continuous flux by discrete gradients.
Keywords: Discrete conservation laws, energy method., discrete gradient methods.
Mathematics Subject Classification: Primary: 35L65, 65M20; Secondary: 65P1.
Citation: Robert I. McLachlan, G. R. W. Quispel. Discrete gradient methods have an energy conservation law. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 1099-1104. doi: 10.3934/dcds.2014.34.1099
References:
[1]

T. J. Bridges and S. Reich, Numerical methods for Hamiltonian PDEs, J. Phys. A, 39 (2006), 5287-5320. doi: 10.1088/0305-4470/39/19/S02.  Google Scholar

[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), 6770-6789. doi: 10.1016/j.jcp.2012.06.022.  Google Scholar

[3]

O. Gonzalez, Time integration and discrete Hamiltonian systems, J. Nonlinear Sci., 6 (1996), 449-467. doi: 10.1007/BF02440162.  Google Scholar

[4]

P. E. Hydon and E. L. Mansfield, A variational complex for difference equations, Found. Comput. Math., 4 (2004) 187-217. doi: 10.1007/s10208-002-0071-9.  Google Scholar

[5]

R. I. McLachlan, G. R. W. Quispel and N. Robidoux, Geometric integration using discrete gradients, Phil. Trans. Roy. Soc. A, 357 (1999), 1021-1045. doi: 10.1098/rsta.1999.0363.  Google Scholar

[6]

M. Oliver and C. Wulff, A-stable Runge-Kutta methods for semilinear evolution equations, J. Funct. Anal., 263 (2012), 1981-2023. doi: 10.1016/j.jfa.2012.06.022.  Google Scholar

[7]

G. R. W. Quispel and D. I. McLaren, A new class of energy-preserving numerical integration methods, J. Phys. A, 41 (2008), 045206, 7pp. doi: 10.1088/1751-8113/41/4/045206.  Google Scholar

[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), L341-L349. doi: 10.1088/0305-4470/29/13/006.  Google Scholar

[9]

E. Rothe, Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben, Math. Ann., 102 (1930), 650-670. doi: 10.1007/BF01782368.  Google Scholar

[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), 847-869. doi: 10.1080/00207160701458633.  Google Scholar

[11]

J. C. Simo and N. Tarnow, The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics, ZAMP, 43 (1992), 757-793. doi: 10.1007/BF00913408.  Google Scholar

[12]

J. Vanneste, On the derivation of fluxes for conservation laws in Hamiltonian systems, IMA J. Appl. Math., 59 (1997), 211-220 doi: 10.1093/imamat/59.2.211.  Google Scholar

show all references

References:
[1]

T. J. Bridges and S. Reich, Numerical methods for Hamiltonian PDEs, J. Phys. A, 39 (2006), 5287-5320. doi: 10.1088/0305-4470/39/19/S02.  Google Scholar

[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), 6770-6789. doi: 10.1016/j.jcp.2012.06.022.  Google Scholar

[3]

O. Gonzalez, Time integration and discrete Hamiltonian systems, J. Nonlinear Sci., 6 (1996), 449-467. doi: 10.1007/BF02440162.  Google Scholar

[4]

P. E. Hydon and E. L. Mansfield, A variational complex for difference equations, Found. Comput. Math., 4 (2004) 187-217. doi: 10.1007/s10208-002-0071-9.  Google Scholar

[5]

R. I. McLachlan, G. R. W. Quispel and N. Robidoux, Geometric integration using discrete gradients, Phil. Trans. Roy. Soc. A, 357 (1999), 1021-1045. doi: 10.1098/rsta.1999.0363.  Google Scholar

[6]

M. Oliver and C. Wulff, A-stable Runge-Kutta methods for semilinear evolution equations, J. Funct. Anal., 263 (2012), 1981-2023. doi: 10.1016/j.jfa.2012.06.022.  Google Scholar

[7]

G. R. W. Quispel and D. I. McLaren, A new class of energy-preserving numerical integration methods, J. Phys. A, 41 (2008), 045206, 7pp. doi: 10.1088/1751-8113/41/4/045206.  Google Scholar

[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), L341-L349. doi: 10.1088/0305-4470/29/13/006.  Google Scholar

[9]

E. Rothe, Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben, Math. Ann., 102 (1930), 650-670. doi: 10.1007/BF01782368.  Google Scholar

[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), 847-869. doi: 10.1080/00207160701458633.  Google Scholar

[11]

J. C. Simo and N. Tarnow, The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics, ZAMP, 43 (1992), 757-793. doi: 10.1007/BF00913408.  Google Scholar

[12]

J. Vanneste, On the derivation of fluxes for conservation laws in Hamiltonian systems, IMA J. Appl. Math., 59 (1997), 211-220 doi: 10.1093/imamat/59.2.211.  Google Scholar

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