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.
Mots-clés : discrete gradient method, energy-preserving integrator, finite difference method, lagrangian mechanics
@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 },
pages = {1493--1513},
publisher = {EDP-Sciences},
volume = {47},
number = {5},
year = {2013},
doi = {10.1051/m2an/2013080},
mrnumber = {3100772},
zbl = {1284.65109},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2013080/}
}
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 PY - 2013 SP - 1493 EP - 1513 VL - 47 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2013080/ DO - 10.1051/m2an/2013080 LA - en ID - M2AN_2013__47_5_1493_0 ER -
%0 Journal Article %A Yaguchi, Takaharu %T Lagrangian approach to deriving energy-preserving numerical schemes for the Euler-Lagrange partial differential equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 1493-1513 %V 47 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2013080/ %R 10.1051/m2an/2013080 %G en %F 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 , Tome 47 (2013) no. 5, pp. 1493-1513. doi : 10.1051/m2an/2013080. http://www.numdam.org/articles/10.1051/m2an/2013080/
[1] and , Foundations of mechanics, 2nd ed. Addison-Wesley (1978). | MR | Zbl
[2] , and , Stabilization of DAEs and invariant manifolds. Numer. Math. 6 (1994) 131-149. | MR | Zbl
[3] , Stabilization of constraints and integrals of motion in dynamical systems. Comput. Math. Appl. Mech. Eng. 1 (1972) 1-16. | MR | Zbl
[4] , and , Precise computations of chemotactic collapse using moving mesh methods. J. Comput. Phys. 202 (2005) 462-487. | MR | Zbl
[5] and , Symmetry adapted moving mesh schemes for the nonlinear Schrodinger equation. J. Phys. A 34 (2001) 10387. | MR | Zbl
[6] , and , Moving mesh methods for problems with blow-up. SIAM J. Sci. Comput. 17 (1996) 305-327. | MR | Zbl
[7] , and , Scaling invariance and adaptivity. Appl. Numer. Math. 39 (2001) 261-288. | MR | Zbl
[8] and , Geometric integration and its applications. in Handbook of Numerical Analysis. North-Holland (2000) 35-139. | MR | Zbl
[9] and , Parabolic Monge-Ampère methods for blow-up problems in several spatial dimensions. J. Phys. A 39 (2006) 5425-5444. | MR | Zbl
[10] and , Moving mesh generation using the parabolic Monge-Ampère equation. SIAM J. Sci. Comput. 31 (2009) 3438-3465. | MR | Zbl
[11] and , How to adaptively resolve evolutionary singularities in differential equations with symmetry. J. Eng. Math. 66 (2010) 217-236. | MR | Zbl
[12] , Discrete calculus of variations. Internat. J. Control 11 (1970) 393-407. | Zbl
[13] , , , , , , and , Preserving energy resp. dissipation in numerical PDEs, using the average vector field method. NTNU reports, Numerics No 7/09. | Zbl
[14] , , , , and , Energy-preserving Runge-Kutta methods. ESAIM: M2AN 43 (2009) 645-649. | Numdam | MR | Zbl
[15] , and , An algebraic approach to invariant preserving integrators: The case of quadratic and Hamiltonian invariants. Numer. Math. 103 (2006) 575-590. | MR | Zbl
[16] and , A general framework for deriving integral preserving numerical methods for PDEs. NTNU reports, Numerics No 8/10. | Zbl
[17] , and , Preserving multiple first integrals by discrete gradients. J. Phys. A 44 (2011) 305205. | MR | Zbl
[18] , Noether-type theorems for difference equations. Appl. Numer. Math. 39 (2001) 307-321. | MR | Zbl
[19] , Applications of Lie Groups to Difference Equations. CRC press, Boca Raton, FL (2010). | MR | Zbl
[20] , Convergence results for a coordinate projection method applied to mechanical systems with algebraic constraints. SIAM J. Numer. Anal. 30 (1993) 1467-1482. | MR | Zbl
[21] and , Symplectic Geometry Algorithms for Hamiltonian Systems. Springer-Verlag, Berlin (2010). | Zbl
[22] , , and , Nonsmooth Lagrangian mechanics and variational collision integrators. SIAM J. Appl. Dynam. Sys. 2 (2003) 381-416. | MR | Zbl
[23] , Finite difference schemes for equation | MR | Zbl
[24] , A stable and conservative finite difference scheme for the Cahn-Hilliard equation. Numer. Math. 87 (2001) 675-699. | MR | Zbl
[25] , Finite difference schemes for nonlinear wave equation that inherit energy conservation property. J. Comput. Appl. Math. 134 (2001) 35-57. | MR | Zbl
[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 | Zbl
[27] and , Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations. CRC Press, Boca Raton, FL (2011). | MR | Zbl
[28] , and , Classical Mechanics, 3rd ed. Addison-Wesley, New York (2002). | MR | Zbl
[29] , Time integration and discrete Hamiltonian systems. J. Nonlinear Sci. 6 (1996) 449-467. | MR | Zbl
[30] , Symmetric projection methods for differential equations on manifolds. BIT 40 (2000) 726-734. | MR | Zbl
[31] , Geometric integration of ordinary differential equations on manifolds. BIT 41 (2001) 996-1007. | MR
[32] , Energy-preserving variant of collocation methods. J. Numer. Anal. Ind. Appl. Math. 5 (2010) 73-84. | MR
[33] , and , Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd ed. Springer-Verlag, Berlin (2006). | MR | Zbl
[34] , and , Moving mesh partial differential equations (MMPDES) based on the equidistribution principle. SIAM J. Numer. Anal. 31 (1994) 709-730. | MR | Zbl
[35] and , A variational complex for difference equations. Found. Comput. Math. 4 (2004) 187-217. | MR | Zbl
[36] and , Hamiltonian-conserving discrete canonical equations based on variational difference quotients. J. Comput. Phys. 76 (1988) 85-102. | MR | Zbl
[37] , , and , Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems. Int. J. Numer. Methods Eng. 49 (2000) 1295-1325. | MR | Zbl
[38] , Solving nonlinear equations with Newton's method. SIAM, Philadelphia (2003). | MR | Zbl
[39] and , Discrete mechanics-a general treatment. J. Comput. Phys. 15 (1974) 134-167. | MR | Zbl
[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 | Zbl
[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 | Zbl
[42] and , Mechanics, 3rd ed. Butterworth-Heinemann, London (1976). | MR
[43] , Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002). | MR | Zbl
[44] , , and , Asynchronous variational integrators. Arch. Ration. Mech. Anal. 167 (2003) 85-146. | MR | Zbl
[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 | Zbl
[46] , First integrals in the discrete variational calculus. Aequationes Math. 9 (1973) 210-220. | MR | Zbl
[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 | Zbl
[48] , and , Multisymplectic geometry, variational integrators, and nonlinear PDEs. Commun. Math. Phys. 199 (1998) 351-395. | MR | Zbl
[49] , , and , Variational methods, multisymplectic geometry and continuum mechanics. J. Geom. Phys. 38 (2001) 253-284. | MR | Zbl
[50] and , Discrete mechanics and variational integrators. Acta Numer. 10 (2001) 357-514. | MR | Zbl
[51] , High-order schemes for conservative or dissipative systems. J. Comput. Appl. Math. 152 (2003) 305-317. | MR | Zbl
[52] , New conservative schemes with discrete variational derivatives for nonlinear wave equations. J. Comput. Appl. Math. 203 (2007) 32-56. | MR | Zbl
[53] , Dissipative/conservative Galerkin method using discrete partial derivative for nonlinear evolution equations. J. Comput. Appl. Math. 218 (2008) 506-521. | MR | Zbl
[54] and , Dissipative or conservative finite difference schemes for complex-valued nonlinear partial differential equations. J. Comput. Phys. 171 (2001) 425-447. | MR | Zbl
[55] , , and , Linearly implicit finite difference schemes derived by the discrete variational method. RIMS Kokyuroku 1145 (2000) 121-129. | MR | Zbl
[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 | Zbl
[57] , and , Geometric integration using discrete gradients. Philos. Trans. Roy. Soc. A 357 (1999) 1021-1046. | MR | Zbl
[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 | Zbl
[60] , Applications of Lie Groups to Differential Equations, 2nd ed. In vol. 107. Graduate Texts in Mathematics. Springer-Verlag, New York (1993). | MR | Zbl
[61] and , On the numerical solution of Euler − Lagrange equations. Mech. Struct. Mach., 19 (1991) 1-18. | MR | Zbl
[62] and , Implicit numerical integration for Euler − Lagrange equations via tangent space parametrization. Mech. Struct. Mach. 19 (1991) 77-98. | MR
[63] and , A new class of energy-preserving numerical integration methods. J. Phys. A 41 (2008) 045206. | MR | Zbl
[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 | Zbl
[66] , Conservation laws and the numerical solution of ODEs. Comput. Math. Appl. B12 (1986) 1287-1296. | MR | Zbl
[67] , Conservation laws and the numerical solution of ODEs II. Comput. Math. Appl. 38 (1999) 61-72. | MR | Zbl
[68] , Variational integrators, Ph.D. thesis, California Institute of Technology (2004). | MR
[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 | Zbl
[70] , and , An extension of the discrete variational method to nonuniform grids. J. Comput. Phys. 229 (2010) 4382-4423. | MR | Zbl
[71] and , Lie-Poisson integrators and Lie-Poisson Hamilton-Jacobi theory. Phys. Lett. A 133 (1988) 134-139. | MR
Cité par Sources :
