# Lagrangian approach to deriving energy-preserving numerical schemes for the Euler–Lagrange partial differential equations

- Volume: 47, Issue: 5, page 1493-1513
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topYaguchi, Takaharu. "Lagrangian approach to deriving energy-preserving numerical schemes for the Euler–Lagrange partial differential equations." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.5 (2013): 1493-1513. <http://eudml.org/doc/273182>.

@article{Yaguchi2013,

abstract = {We propose a Lagrangian approach to deriving energy-preserving finite difference schemes for the Euler–Lagrange partial differential equations. Noether’s theorem states that the symmetry of time translation of Lagrangians yields the energy conservation law. We introduce a unique viewpoint on this theorem: “the symmetry of time translation of Lagrangians derives the Euler–Lagrange equation and the energy conservation law, simultaneously.” The proposed method is a combination of a discrete counter part of this statement and the discrete gradient method. It is also shown that the symmetry of space translation derives momentum-preserving schemes. Finally, we discuss the existence of discrete local conservation laws.},

author = {Yaguchi, Takaharu},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {discrete gradient method; energy-preserving integrator; finite difference method; lagrangian mechanics; Lagrangian mechanics; Hamilton-Jacobi equation; conservation law},

language = {eng},

number = {5},

pages = {1493-1513},

publisher = {EDP-Sciences},

title = {Lagrangian approach to deriving energy-preserving numerical schemes for the Euler–Lagrange partial differential equations},

url = {http://eudml.org/doc/273182},

volume = {47},

year = {2013},

}

TY - JOUR

AU - Yaguchi, Takaharu

TI - Lagrangian approach to deriving energy-preserving numerical schemes for the Euler–Lagrange partial differential equations

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2013

PB - EDP-Sciences

VL - 47

IS - 5

SP - 1493

EP - 1513

AB - We propose a Lagrangian approach to deriving energy-preserving finite difference schemes for the Euler–Lagrange partial differential equations. Noether’s theorem states that the symmetry of time translation of Lagrangians yields the energy conservation law. We introduce a unique viewpoint on this theorem: “the symmetry of time translation of Lagrangians derives the Euler–Lagrange equation and the energy conservation law, simultaneously.” The proposed method is a combination of a discrete counter part of this statement and the discrete gradient method. It is also shown that the symmetry of space translation derives momentum-preserving schemes. Finally, we discuss the existence of discrete local conservation laws.

LA - eng

KW - discrete gradient method; energy-preserving integrator; finite difference method; lagrangian mechanics; Lagrangian mechanics; Hamilton-Jacobi equation; conservation law

UR - http://eudml.org/doc/273182

ER -

## References

top- [1] R. Abraham and J.E. Marsden, Foundations of mechanics, 2nd ed. Addison-Wesley (1978). Zbl0393.70001MR515141
- [2] U.M. Ascher, H. Chin and S. Reich, Stabilization of DAEs and invariant manifolds. Numer. Math.6 (1994) 131–149. Zbl0791.65051MR1262777
- [3] J. Baumgarte, Stabilization of constraints and integrals of motion in dynamical systems. Comput. Math. Appl. Mech. Eng.1 (1972) 1–16. Zbl0262.70017MR391628
- [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. Zbl1063.65096MR2145389
- [5] C.J. Budd and V. Dorodnitsyn, Symmetry adapted moving mesh schemes for the nonlinear Schrodinger equation. J. Phys. A 34 (2001) 10387. Zbl0991.65085MR1877461
- [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. Zbl0860.35050MR1374282
- [7] C.J. Budd, B. Leimkuhler and M.D. Piggott, Scaling invariance and adaptivity. Appl. Numer. Math.39 (2001) 261–288. Zbl0998.65070MR1866591
- [8] C.J. Budd and M.D. Piggott, Geometric integration and its applications. in Handbook of Numerical Analysis. North-Holland (2000) 35–139. Zbl1062.65134MR2009771
- [9] C.J. Budd and J.F. Williams, Parabolic Monge-Ampère methods for blow-up problems in several spatial dimensions. J. Phys. A39 (2006) 5425–5444. Zbl1096.35070MR2220768
- [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. Zbl1200.65099MR2538864
- [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. Zbl1204.35016MR2585823
- [12] J.A. Cadzow, Discrete calculus of variations. Internat. J. Control11 (1970) 393–407. Zbl0193.07601
- [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. Zbl1284.65184
- [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. Zbl1169.65348MR2542869
- [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. Zbl1100.65115MR2221062
- [16] M. Dahlby and B. Owren, A general framework for deriving integral preserving numerical methods for PDEs. NTNU reports, Numerics No 8/10. Zbl1246.65240
- [17] M. Dahlby, B. Owren and T. Yaguchi, Preserving multiple first integrals by discrete gradients. J. Phys. A 44 (2011) 305205. Zbl1245.65174MR2817743
- [18] V. Dorodnitsyn, Noether-type theorems for difference equations. Appl. Numer. Math.39 (2001) 307–321. Zbl0995.39002MR1866593
- [19] V. Dorodnitsyn, Applications of Lie Groups to Difference Equations. CRC press, Boca Raton, FL (2010). Zbl1236.39002MR2759751
- [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. Zbl0785.65079MR1239831
- [21] K. Feng and M. Qin, Symplectic Geometry Algorithms for Hamiltonian Systems. Springer-Verlag, Berlin (2010). Zbl1207.65149
- [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. Zbl1088.37045MR2031279
- [23] D. Furihata, Finite difference schemes for equation$\frac{\partial u}{\partial t}={\left(\frac{\partial}{\partial x}\right)}^{\alpha}\frac{\delta G}{\delta u}$∂u∂t=∂∂x)(αδGδu that inherit energy conservation or dissipation property. J. Comput. Phys. 156 (1999) 181–205. Zbl0945.65103MR1727636
- [24] D. Furihata, A stable and conservative finite difference scheme for the Cahn–Hilliard equation. Numer. Math.87 (2001) 675–699. Zbl0974.65086MR1815731
- [25] D. Furihata, Finite difference schemes for nonlinear wave equation that inherit energy conservation property. J. Comput. Appl. Math.134 (2001) 35–57. Zbl0989.65099MR1852556
- [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. Zbl1035.65100MR1956298
- [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). Zbl1227.65094MR2744841
- [28] H. Goldstein, C. Poole and J. Safko, Classical Mechanics, 3rd ed. Addison-Wesley, New York (2002). Zbl1132.70001MR43608
- [29] O. Gonzalez, Time integration and discrete Hamiltonian systems. J. Nonlinear Sci.6 (1996) 449–467. Zbl0866.58030MR1411343
- [30] E. Hairer, Symmetric projection methods for differential equations on manifolds. BIT40 (2000) 726–734. Zbl0968.65108MR1799312
- [31] E. Hairer, Geometric integration of ordinary differential equations on manifolds. BIT41 (2001) 996–1007. MR2058858
- [32] E. Hairer, Energy-preserving variant of collocation methods. J. Numer. Anal. Ind. Appl. Math.5 (2010) 73–84. MR2833602
- [33] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd ed. Springer-Verlag, Berlin (2006). Zbl0994.65135MR2221614
- [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. Zbl0806.65092MR1275109
- [35] P.E. Hydon and E.L. Mansfield, A variational complex for difference equations. Found. Comput. Math.4 (2004) 187–217. Zbl1057.39013MR2049870
- [36] T. Itoh and K. Abe, Hamiltonian-conserving discrete canonical equations based on variational difference quotients. J. Comput. Phys.76 (1988) 85–102. Zbl0656.70015MR943488
- [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. Zbl0969.70004MR1805500
- [38] C.T. Kelley, Solving nonlinear equations with Newton’s method. SIAM, Philadelphia (2003). Zbl1031.65069MR1998383
- [39] R.A. LaBudde and D. Greenspan, Discrete mechanics—a general treatment. J. Comput. Phys.15 (1974) 134–167. Zbl0301.70006MR351213
- [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. Zbl0364.65066MR433889
- [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. Zbl0382.65031MR445980
- [42] L.D. Landau and E.M. Lifshitz, Mechanics, 3rd ed. Butterworth-Heinemann, London (1976). MR475051
- [43] R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002). Zbl1010.65040MR1925043
- [44] A. Lew, J.E. Marsden, M. Ortiz and M. West, Asynchronous variational integrators. Arch. Ration. Mech. Anal.167 (2003) 85–146. Zbl1055.74041MR1971150
- [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. Zbl0847.65062MR1360462
- [46] J.D. Logan, First integrals in the discrete variational calculus. Aequationes Math.9 (1973) 210–220. Zbl0268.49022MR328397
- [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. Zbl1080.65054MR2182821
- [48] J.E. Marsden, G.W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs. Commun. Math. Phys.199 (1998) 351-395. Zbl0951.70002MR1666871
- [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. Zbl1007.74018MR1829044
- [50] J.E. Marsden and M. West, Discrete mechanics and variational integrators. Acta Numer.10 (2001) 357–514. Zbl1123.37327MR2009697
- [51] T. Matsuo, High-order schemes for conservative or dissipative systems. J. Comput. Appl. Math.152 (2003) 305–317. Zbl1019.65042MR1991298
- [52] T. Matsuo, New conservative schemes with discrete variational derivatives for nonlinear wave equations. J. Comput. Appl. Math.203 (2007) 32–56. Zbl1120.65096MR2313820
- [53] T. Matsuo, Dissipative/conservative Galerkin method using discrete partial derivative for nonlinear evolution equations. J. Comput. Appl. Math.218 (2008) 506–521. Zbl1147.65078MR2437123
- [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. Zbl0993.65098MR1848726
- [55] T. Matsuo, M. Sugihara, D. Furihata and M. Mori, Linearly implicit finite difference schemes derived by the discrete variational method. RIMS Kokyuroku1145 (2000) 121–129. Zbl0968.65521MR1795452
- [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. Zbl1014.65083MR1933890
- [57] R.I. McLachlan, G.R.W. Quispel and N. Robidoux, Geometric integration using discrete gradients. Philos. Trans. Roy. Soc. A357 (1999) 1021–1046. Zbl0933.65143MR1694701
- [58] R.I. McLachlan and N. Robidoux, Antisymmetry, pseudospectral methods, weighted residual discretizations, and energy conserving partial differential equations, preprint. Zbl0963.65105
- [59] K.S. Miller, Linear difference equations, W.A. Benjamin Inc., New York–Amsterdam (1968). Zbl0162.40201MR227644
- [60] P. Olver, Applications of Lie Groups to Differential Equations, 2nd ed. In vol. 107. Graduate Texts in Mathematics. Springer-Verlag, New York (1993). Zbl0785.58003MR1240056
- [61] F.A. Potra and W.C. Rheinboldt, On the numerical solution of Euler − Lagrange equations. Mech. Struct. Mach., 19 (1991) 1–18. Zbl0818.65064MR1142054
- [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. MR1142057
- [63] G.R.W. Quispel and D.I. McLaren, A new class of energy-preserving numerical integration methods. J. Phys. A 41 (2008) 045206. Zbl1132.65065MR2451073
- [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.
- [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). Zbl0816.65042MR1270017
- [66] L.F. Shampine, Conservation laws and the numerical solution of ODEs. Comput. Math. Appl. B12 (1986) 1287–1296. Zbl0641.65057MR871366
- [67] L.F. Shampine, Conservation laws and the numerical solution of ODEs II. Comput. Math. Appl.38 (1999) 61–72. Zbl0947.65086MR1698919
- [68] M. West, Variational integrators, Ph.D. thesis, California Institute of Technology (2004). Zbl1060.70500MR2706618
- [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. Zbl0963.65537MR1870276
- [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. Zbl1190.65128MR2609783
- [71] G. Zhong and J.E. Marsden, Lie–Poisson integrators and Lie–Poisson Hamilton–Jacobi theory. Phys. Lett. A133 (1988) 134–139. MR967725

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.