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.
@article{M2AN_2013__47_5_1493_0,
author = {Yaguchi, Takaharu},
title = {Lagrangian approach to deriving energy-preserving numerical schemes for the Euler-Lagrange partial differential equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {47},
year = {2013},
pages = {1493-1513},
doi = {10.1051/m2an/2013080},
mrnumber = {3100772},
zbl = {1284.65109},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2013__47_5_1493_0}
}
Yaguchi, 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, Tome 47 (2013) pp. 1493-1513. doi : 10.1051/m2an/2013080. http://gdmltest.u-ga.fr/item/M2AN_2013__47_5_1493_0/
[1] and , Foundations of mechanics, 2nd ed. Addison-Wesley (1978). | MR 515141 | Zbl 0393.70001
[2] , and , Stabilization of DAEs and invariant manifolds. Numer. Math. 6 (1994) 131-149. | MR 1262777 | Zbl 0791.65051
[3] , Stabilization of constraints and integrals of motion in dynamical systems. Comput. Math. Appl. Mech. Eng. 1 (1972) 1-16. | MR 391628 | Zbl 0262.70017
[4] , and , Precise computations of chemotactic collapse using moving mesh methods. J. Comput. Phys. 202 (2005) 462-487. | MR 2145389 | Zbl 1063.65096
[5] and , Symmetry adapted moving mesh schemes for the nonlinear Schrodinger equation. J. Phys. A 34 (2001) 10387. | MR 1877461 | Zbl 0991.65085
[6] , and , Moving mesh methods for problems with blow-up. SIAM J. Sci. Comput. 17 (1996) 305-327. | MR 1374282 | Zbl 0860.35050
[7] , and , Scaling invariance and adaptivity. Appl. Numer. Math. 39 (2001) 261-288. | MR 1866591 | Zbl 0998.65070
[8] and , Geometric integration and its applications. in Handbook of Numerical Analysis. North-Holland (2000) 35-139. | MR 2009771 | Zbl 1062.65134
[9] and , Parabolic Monge-Ampère methods for blow-up problems in several spatial dimensions. J. Phys. A 39 (2006) 5425-5444. | MR 2220768 | Zbl 1096.35070
[10] and , Moving mesh generation using the parabolic Monge-Ampère equation. SIAM J. Sci. Comput. 31 (2009) 3438-3465. | MR 2538864 | Zbl 1200.65099
[11] and , How to adaptively resolve evolutionary singularities in differential equations with symmetry. J. Eng. Math. 66 (2010) 217-236. | MR 2585823 | Zbl 1204.35016
[12] , Discrete calculus of variations. Internat. J. Control 11 (1970) 393-407. | Zbl 0193.07601
[13] , , , , , , and , Preserving energy resp. dissipation in numerical PDEs, using the average vector field method. NTNU reports, Numerics No 7/09. | Zbl 1284.65184
[14] , , , , and , Energy-preserving Runge-Kutta methods. ESAIM: M2AN 43 (2009) 645-649. | Numdam | MR 2542869 | Zbl 1169.65348
[15] , and , An algebraic approach to invariant preserving integrators: The case of quadratic and Hamiltonian invariants. Numer. Math. 103 (2006) 575-590. | MR 2221062 | Zbl 1100.65115
[16] and , A general framework for deriving integral preserving numerical methods for PDEs. NTNU reports, Numerics No 8/10. | Zbl 1246.65240
[17] , and , Preserving multiple first integrals by discrete gradients. J. Phys. A 44 (2011) 305205. | MR 2817743 | Zbl 1245.65174
[18] , Noether-type theorems for difference equations. Appl. Numer. Math. 39 (2001) 307-321. | MR 1866593 | Zbl 0995.39002
[19] , Applications of Lie Groups to Difference Equations. CRC press, Boca Raton, FL (2010). | MR 2759751 | Zbl 1236.39002
[20] , Convergence results for a coordinate projection method applied to mechanical systems with algebraic constraints. SIAM J. Numer. Anal. 30 (1993) 1467-1482. | MR 1239831 | Zbl 0785.65079
[21] and , Symplectic Geometry Algorithms for Hamiltonian Systems. Springer-Verlag, Berlin (2010). | Zbl 1207.65149
[22] , , and , Nonsmooth Lagrangian mechanics and variational collision integrators. SIAM J. Appl. Dynam. Sys. 2 (2003) 381-416. | MR 2031279 | Zbl 1088.37045
[23] , Finite difference schemes for equation | MR 1727636 | Zbl 0945.65103
[24] , A stable and conservative finite difference scheme for the Cahn-Hilliard equation. Numer. Math. 87 (2001) 675-699. | MR 1815731 | Zbl 0974.65086
[25] , Finite difference schemes for nonlinear wave equation that inherit energy conservation property. J. Comput. Appl. Math. 134 (2001) 35-57. | MR 1852556 | Zbl 0989.65099
[26] and , A Stable, convergent, conservative and linear finite difference scheme for the Cahn-Hilliard equation. Japan J. Indust. Appl. Math. 20 (2003) 65-85. | MR 1956298 | Zbl 1035.65100
[27] and , Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations. CRC Press, Boca Raton, FL (2011). | MR 2744841 | Zbl 1227.65094
[28] , and , Classical Mechanics, 3rd ed. Addison-Wesley, New York (2002). | MR 43608 | Zbl 1132.70001
[29] , Time integration and discrete Hamiltonian systems. J. Nonlinear Sci. 6 (1996) 449-467. | MR 1411343 | Zbl 0866.58030
[30] , Symmetric projection methods for differential equations on manifolds. BIT 40 (2000) 726-734. | MR 1799312 | Zbl 0968.65108
[31] , Geometric integration of ordinary differential equations on manifolds. BIT 41 (2001) 996-1007. | MR 2058858
[32] , Energy-preserving variant of collocation methods. J. Numer. Anal. Ind. Appl. Math. 5 (2010) 73-84. | MR 2833602
[33] , and , Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd ed. Springer-Verlag, Berlin (2006). | MR 2221614 | Zbl 0994.65135
[34] , and , Moving mesh partial differential equations (MMPDES) based on the equidistribution principle. SIAM J. Numer. Anal. 31 (1994) 709-730. | MR 1275109 | Zbl 0806.65092
[35] and , A variational complex for difference equations. Found. Comput. Math. 4 (2004) 187-217. | MR 2049870 | Zbl 1057.39013
[36] and , Hamiltonian-conserving discrete canonical equations based on variational difference quotients. J. Comput. Phys. 76 (1988) 85-102. | MR 943488 | Zbl 0656.70015
[37] , , and , Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems. Int. J. Numer. Methods Eng. 49 (2000) 1295-1325. | MR 1805500 | Zbl 0969.70004
[38] , Solving nonlinear equations with Newton's method. SIAM, Philadelphia (2003). | MR 1998383 | Zbl 1031.65069
[39] and , Discrete mechanics-a general treatment. J. Comput. Phys. 15 (1974) 134-167. | MR 351213 | Zbl 0301.70006
[40] and , 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. | MR 433889 | Zbl 0364.65066
[41] and , 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. | MR 445980 | Zbl 0382.65031
[42] and , Mechanics, 3rd ed. Butterworth-Heinemann, London (1976). | MR 475051
[43] , Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002). | MR 1925043 | Zbl 1010.65040
[44] , , and , Asynchronous variational integrators. Arch. Ration. Mech. Anal. 167 (2003) 85-146. | MR 1971150 | Zbl 1055.74041
[45] and , Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation. SIAM J. Numer. Anal. 32 (1995) 1839-1875. | MR 1360462 | Zbl 0847.65062
[46] , First integrals in the discrete variational calculus. Aequationes Math. 9 (1973) 210-220. | MR 328397 | Zbl 0268.49022
[47] and , 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. | MR 2182821 | Zbl 1080.65054
[48] , and , Multisymplectic geometry, variational integrators, and nonlinear PDEs. Commun. Math. Phys. 199 (1998) 351-395. | MR 1666871 | Zbl 0951.70002
[49] , , and , Variational methods, multisymplectic geometry and continuum mechanics. J. Geom. Phys. 38 (2001) 253-284. | MR 1829044 | Zbl 1007.74018
[50] and , Discrete mechanics and variational integrators. Acta Numer. 10 (2001) 357-514. | MR 2009697 | Zbl 1123.37327
[51] , High-order schemes for conservative or dissipative systems. J. Comput. Appl. Math. 152 (2003) 305-317. | MR 1991298 | Zbl 1019.65042
[52] , New conservative schemes with discrete variational derivatives for nonlinear wave equations. J. Comput. Appl. Math. 203 (2007) 32-56. | MR 2313820 | Zbl 1120.65096
[53] , Dissipative/conservative Galerkin method using discrete partial derivative for nonlinear evolution equations. J. Comput. Appl. Math. 218 (2008) 506-521. | MR 2437123 | Zbl 1147.65078
[54] and , Dissipative or conservative finite difference schemes for complex-valued nonlinear partial differential equations. J. Comput. Phys. 171 (2001) 425-447. | MR 1848726 | Zbl 0993.65098
[55] , , and , Linearly implicit finite difference schemes derived by the discrete variational method. RIMS Kokyuroku 1145 (2000) 121-129. | MR 1795452 | Zbl 0968.65521
[56] , , and , Spatially accurate dissipative or conservative finite difference schemes derived by the discrete variational method. Japan J. Indust. Appl. Math. 19 (2002) 311-330. | MR 1933890 | Zbl 1014.65083
[57] , and , Geometric integration using discrete gradients. Philos. Trans. Roy. Soc. A 357 (1999) 1021-1046. | MR 1694701 | Zbl 0933.65143
[58] and , Antisymmetry, pseudospectral methods, weighted residual discretizations, and energy conserving partial differential equations, preprint.
[59] , Linear difference equations, W.A. Benjamin Inc., New York-Amsterdam (1968). | MR 227644 | Zbl 0162.40201
[60] , Applications of Lie Groups to Differential Equations, 2nd ed. In vol. 107. Graduate Texts in Mathematics. Springer-Verlag, New York (1993). | MR 1240056 | Zbl 0785.58003
[61] and , On the numerical solution of Euler − Lagrange equations. Mech. Struct. Mach., 19 (1991) 1-18. | MR 1142054 | Zbl 0818.65064
[62] and , Implicit numerical integration for Euler − Lagrange equations via tangent space parametrization. Mech. Struct. Mach. 19 (1991) 77-98. | MR 1142057
[63] and , A new class of energy-preserving numerical integration methods. J. Phys. A 41 (2008) 045206. | MR 2451073 | Zbl 1132.65065
[64] , Symplectic finite difference time domain methods for Maxwell equations -formulation and their properties-. In Book of Abstracts of SciCADE 2009 (2009) 183.
[65] and , Numerical Hamiltonian Problems. In vol. 7 of Applied Mathematics and Mathematical Computation. Chapman and Hall, London (1994). | MR 1270017 | Zbl 0816.65042
[66] , Conservation laws and the numerical solution of ODEs. Comput. Math. Appl. B12 (1986) 1287-1296. | MR 871366 | Zbl 0641.65057
[67] , Conservation laws and the numerical solution of ODEs II. Comput. Math. Appl. 38 (1999) 61-72. | MR 1698919 | Zbl 0947.65086
[68] , Variational integrators, Ph.D. thesis, California Institute of Technology (2004). | MR 2706618
[69] , , and , 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. | MR 1870276 | Zbl 0963.65537
[70] , and , An extension of the discrete variational method to nonuniform grids. J. Comput. Phys. 229 (2010) 4382-4423. | MR 2609783 | Zbl 1190.65128
[71] and , Lie-Poisson integrators and Lie-Poisson Hamilton-Jacobi theory. Phys. Lett. A 133 (1988) 134-139. | MR 967725