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

Published online by Cambridge University Press:  01 August 2013

Takaharu Yaguchi
Takaharu Yaguchi*
Affiliation:
Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe, 657-8501, Japan
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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.

Keywords

Discrete gradient methodenergy-preserving integratorfinite difference methodLagrangian mechanics
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 5 , September 2013 , pp. 1493 - 1513
Copyright
© EDP Sciences, SMAI, 2013

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References

R. Abraham and J.E. Marsden, Foundations of mechanics, 2nd ed. Addison-Wesley (1978).
Ascher, U.M., Chin, H. and Reich, S., Stabilization of DAEs and invariant manifolds. Numer. Math. 6 (1994) 131–149. Google Scholar
Baumgarte, J., Stabilization of constraints and integrals of motion in dynamical systems. Comput. Math. Appl. Mech. Eng. 1 (1972) 1–16. Google Scholar
Budd, C.J., Carretero-Gonzalez, R. and Russell, R.D., Precise computations of chemotactic collapse using moving mesh methods. J. Comput. Phys. 202 (2005) 462–487. Google Scholar
Budd, C.J. and Dorodnitsyn, V., Symmetry adapted moving mesh schemes for the nonlinear Schrodinger equation. J. Phys. A 34 (2001) 10387. Google Scholar
Budd, C.J., Huang, W.Z. and Russell, R.D., Moving mesh methods for problems with blow-up. SIAM J. Sci. Comput. 17 (1996) 305–327. Google Scholar
Budd, C.J., Leimkuhler, B. and Piggott, M.D., Scaling invariance and adaptivity. Appl. Numer. Math. 39 (2001) 261–288. Google Scholar
C.J. Budd and M.D. Piggott, Geometric integration and its applications. in Handbook of Numerical Analysis. North-Holland (2000) 35–139.
Budd, C.J. and Williams, J.F., Parabolic Monge-Ampère methods for blow-up problems in several spatial dimensions. J. Phys. A 39 (2006) 5425–5444. Google Scholar
Budd, C.J. and Williams, J.F., Moving mesh generation using the parabolic Monge-Ampère equation. SIAM J. Sci. Comput. 31 (2009) 3438–3465. Google Scholar
Budd, C.J. and Williams, J.F., How to adaptively resolve evolutionary singularities in differential equations with symmetry. J. Eng. Math. 66 (2010) 217–236. Google Scholar
Cadzow, J.A., Discrete calculus of variations. Internat. J. Control 11 (1970) 393–407. Google Scholar
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.
Celledoni, E., McLachlan, R.I., McLaren, D.I., Owren, B., Quispel, G.R.W. and Wright, W.M., Energy-preserving Runge–Kutta methods. ESAIM: M2AN 43 (2009) 645–649. Google Scholar
Chartier, P., Faou, E. and Murua, A., An algebraic approach to invariant preserving integrators: The case of quadratic and Hamiltonian invariants. Numer. Math. 103 (2006) 575–590. Google Scholar
M. Dahlby and B. Owren, A general framework for deriving integral preserving numerical methods for PDEs. NTNU reports, Numerics No 8/10.
Dahlby, M., Owren, B. and Yaguchi, T., Preserving multiple first integrals by discrete gradients. J. Phys. A 44 (2011) 305205. Google Scholar
Dorodnitsyn, V., Noether-type theorems for difference equations. Appl. Numer. Math. 39 (2001) 307–321. Google Scholar
V. Dorodnitsyn, Applications of Lie Groups to Difference Equations. CRC press, Boca Raton, FL (2010).
Eich, E., Convergence results for a coordinate projection method applied to mechanical systems with algebraic constraints. SIAM J. Numer. Anal. 30 (1993) 1467–1482. Google Scholar
K. Feng and M. Qin, Symplectic Geometry Algorithms for Hamiltonian Systems. Springer-Verlag, Berlin (2010).
Fetecau, R.C., Marsden, J.E., Ortiz, M. and West, M., Nonsmooth Lagrangian mechanics and variational collision integrators. SIAM J. Appl. Dynam. Sys. 2 (2003) 381–416. Google Scholar
Furihata, D., Finite difference schemes for equation \hbox{$\frac{\partial u}{\partial t} = \left(\frac{\partial }{\partial x}\right)^\alpha \frac{\delta G}{\delta u}$}∂u∂t=∂∂x)(αδGδu that inherit energy conservation or dissipation property. J. Comput. Phys. 156 (1999) 181–205. Google Scholar
Furihata, D., A stable and conservative finite difference scheme for the Cahn–Hilliard equation. Numer. Math. 87 (2001) 675–699. Google Scholar
Furihata, D., Finite difference schemes for nonlinear wave equation that inherit energy conservation property. J. Comput. Appl. Math. 134 (2001) 35–57. Google Scholar
Furihata, D. and Matsuo, T., A Stable, convergent, conservative and linear finite difference scheme for the Cahn–Hilliard equation. Japan J. Indust. Appl. Math. 20 (2003) 65–85. Google Scholar
D. Furihata and T. Matsuo, Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations. CRC Press, Boca Raton, FL (2011).
H. Goldstein, C. Poole and J. Safko, Classical Mechanics, 3rd ed. Addison-Wesley, New York (2002).
Gonzalez, O., Time integration and discrete Hamiltonian systems. J. Nonlinear Sci. 6 (1996) 449–467. Google Scholar
Hairer, E., Symmetric projection methods for differential equations on manifolds. BIT 40 (2000) 726–734. Google Scholar
Hairer, E., Geometric integration of ordinary differential equations on manifolds. BIT 41 (2001) 996–1007. Google Scholar
Hairer, E., Energy-preserving variant of collocation methods. J. Numer. Anal. Ind. Appl. Math. 5 (2010) 73–84. Google Scholar
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd ed. Springer-Verlag, Berlin (2006).
Huang, W., Ren, Y. and Russell, R.D., Moving mesh partial differential equations (MMPDES) based on the equidistribution principle. SIAM J. Numer. Anal. 31 (1994) 709–730. Google Scholar
Hydon, P.E. and Mansfield, E.L., A variational complex for difference equations. Found. Comput. Math. 4 (2004) 187–217. Google Scholar
Itoh, T. and Abe, K., Hamiltonian-conserving discrete canonical equations based on variational difference quotients. J. Comput. Phys. 76 (1988) 85–102. Google Scholar
Kane, C., Marsden, J.E., Ortiz, M. and West, M., Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems. Int. J. Numer. Methods Eng. 49 (2000) 1295–1325. Google Scholar
C.T. Kelley, Solving nonlinear equations with Newton’s method. SIAM, Philadelphia (2003).
LaBudde, R.A. and Greenspan, D., Discrete mechanics—a general treatment. J. Comput. Phys. 15 (1974) 134–167. Google Scholar
LaBudde, R.A. and Greenspan, D., 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. Google Scholar
LaBudde, R.A. and Greenspan, D., 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. Google Scholar
L.D. Landau and E.M. Lifshitz, Mechanics, 3rd ed. Butterworth-Heinemann, London (1976).
R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002).
Lew, A., Marsden, J.E., Ortiz, M. and West, M., Asynchronous variational integrators. Arch. Ration. Mech. Anal. 167 (2003) 85–146. Google Scholar
Li, S. and Vu-Quoc, L., Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein–Gordon equation. SIAM J. Numer. Anal. 32 (1995) 1839–1875. Google Scholar
Logan, J.D., First integrals in the discrete variational calculus. Aequationes Math. 9 (1973) 210–220. Google Scholar
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.
Marsden, J.E., Patrick, G.W. and Shkoller, S., Multisymplectic geometry, variational integrators, and nonlinear PDEs. Commun. Math. Phys. 199 (1998) 351-395. Google Scholar
Marsden, J.E., Pekarsky, S., Shkoller, S. and West, M., Variational methods, multisymplectic geometry and continuum mechanics. J. Geom. Phys. 38 (2001) 253–284. Google Scholar
Marsden, J.E. and West, M., Discrete mechanics and variational integrators. Acta Numer. 10 (2001) 357–514. Google Scholar
Matsuo, T., High-order schemes for conservative or dissipative systems. J. Comput. Appl. Math. 152 (2003) 305–317. Google Scholar
Matsuo, T., New conservative schemes with discrete variational derivatives for nonlinear wave equations. J. Comput. Appl. Math. 203 (2007) 32–56. Google Scholar
Matsuo, T., Dissipative/conservative Galerkin method using discrete partial derivative for nonlinear evolution equations. J. Comput. Appl. Math. 218 (2008) 506–521. Google Scholar
Matsuo, T. and Furihata, D., Dissipative or conservative finite difference schemes for complex-valued nonlinear partial differential equations. J. Comput. Phys. 171 (2001) 425–447. Google Scholar
Matsuo, T., Sugihara, M., Furihata, D. and Mori, M., Linearly implicit finite difference schemes derived by the discrete variational method. RIMS Kokyuroku 1145 (2000) 121–129. Google Scholar
Matsuo, T., Sugihara, M., Furihata, D. and Mori, M., Spatially accurate dissipative or conservative finite difference schemes derived by the discrete variational method. Japan J. Indust. Appl. Math. 19 (2002) 311–330. Google Scholar
McLachlan, R.I., Quispel, G.R.W. and Robidoux, N., Geometric integration using discrete gradients. Philos. Trans. Roy. Soc. A 357 (1999) 1021–1046. Google Scholar
R.I. McLachlan and N. Robidoux, Antisymmetry, pseudospectral methods, weighted residual discretizations, and energy conserving partial differential equations, preprint.
K.S. Miller, Linear difference equations, W.A. Benjamin Inc., New York–Amsterdam (1968).
P. Olver, Applications of Lie Groups to Differential Equations, 2nd ed. In vol. 107. Graduate Texts in Mathematics. Springer-Verlag, New York (1993).
Potra, F.A. and Rheinboldt, W.C., On the numerical solution of Euler − Lagrange equations. Mech. Struct. Mach., 19 (1991) 1–18. Google Scholar
Potra, F.A. and Yen, J., Implicit numerical integration for Euler − Lagrange equations via tangent space parametrization. Mech. Struct. Mach. 19 (1991) 77–98. Google Scholar
Quispel, G.R.W. and McLaren, D.I., A new class of energy-preserving numerical integration methods. J. Phys. A 41 (2008) 045206. Google Scholar
I. Saitoh, Symplectic finite difference time domain methods for Maxwell equations -formulation and their properties-. In Book of Abstracts of SciCADE 2009 (2009) 183.
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).
Shampine, L.F., Conservation laws and the numerical solution of ODEs. Comput. Math. Appl. B12 (1986) 1287–1296. Google Scholar
Shampine, L.F., Conservation laws and the numerical solution of ODEs II. Comput. Math. Appl. 38 (1999) 61–72. Google Scholar
M. West, Variational integrators, Ph.D. thesis, California Institute of Technology (2004).
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.
Yaguchi, T., Matsuo, T. and Sugihara, M., An extension of the discrete variational method to nonuniform grids. J. Comput. Phys. 229 (2010) 4382–4423. Google Scholar
Zhong, G. and Marsden, J.E., Lie–Poisson integrators and Lie–Poisson Hamilton–Jacobi theory. Phys. Lett. A 133 (1988) 134–139. Google Scholar