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

Yaguchi, Takaharu

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 5, pp. 1493-1513.

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.

[1] R. Abraham and J.E. Marsden, Foundations of mechanics, 2nd ed. Addison-Wesley (1978). | MR | Zbl

[2] U.M. Ascher, H. Chin and S. Reich, Stabilization of DAEs and invariant manifolds. Numer. Math. 6 (1994) 131-149. | MR | Zbl

[3] J. Baumgarte, Stabilization of constraints and integrals of motion in dynamical systems. Comput. Math. Appl. Mech. Eng. 1 (1972) 1-16. | MR | Zbl

[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. | MR | Zbl

[5] C.J. Budd and V. Dorodnitsyn, Symmetry adapted moving mesh schemes for the nonlinear Schrodinger equation. J. Phys. A 34 (2001) 10387. | MR | Zbl

[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. | MR | Zbl

[7] C.J. Budd, B. Leimkuhler and M.D. Piggott, Scaling invariance and adaptivity. Appl. Numer. Math. 39 (2001) 261-288. | MR | Zbl

[8] C.J. Budd and M.D. Piggott, Geometric integration and its applications. in Handbook of Numerical Analysis. North-Holland (2000) 35-139. | MR | Zbl

[9] C.J. Budd and J.F. Williams, Parabolic Monge-Ampère methods for blow-up problems in several spatial dimensions. J. Phys. A 39 (2006) 5425-5444. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[12] J.A. Cadzow, Discrete calculus of variations. Internat. J. Control 11 (1970) 393-407. | Zbl

[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. | Zbl

[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. | Numdam | MR | Zbl

[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. | MR | Zbl

[16] M. Dahlby and B. Owren, A general framework for deriving integral preserving numerical methods for PDEs. NTNU reports, Numerics No 8/10. | Zbl

[17] M. Dahlby, B. Owren and T. Yaguchi, Preserving multiple first integrals by discrete gradients. J. Phys. A 44 (2011) 305205. | MR | Zbl

[18] V. Dorodnitsyn, Noether-type theorems for difference equations. Appl. Numer. Math. 39 (2001) 307-321. | MR | Zbl

[19] V. Dorodnitsyn, Applications of Lie Groups to Difference Equations. CRC press, Boca Raton, FL (2010). | MR | Zbl

[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. | MR | Zbl

[21] K. Feng and M. Qin, Symplectic Geometry Algorithms for Hamiltonian Systems. Springer-Verlag, Berlin (2010). | Zbl

[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. | MR | Zbl

[23] D. Furihata, Finite difference schemes for equation | MR | Zbl

[24] D. Furihata, A stable and conservative finite difference scheme for the Cahn-Hilliard equation. Numer. Math. 87 (2001) 675-699. | MR | Zbl

[25] D. Furihata, Finite difference schemes for nonlinear wave equation that inherit energy conservation property. J. Comput. Appl. Math. 134 (2001) 35-57. | MR | Zbl

[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. | MR | Zbl

[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). | MR | Zbl

[28] H. Goldstein, C. Poole and J. Safko, Classical Mechanics, 3rd ed. Addison-Wesley, New York (2002). | MR | Zbl

[29] O. Gonzalez, Time integration and discrete Hamiltonian systems. J. Nonlinear Sci. 6 (1996) 449-467. | MR | Zbl

[30] E. Hairer, Symmetric projection methods for differential equations on manifolds. BIT 40 (2000) 726-734. | MR | Zbl

[31] E. Hairer, Geometric integration of ordinary differential equations on manifolds. BIT 41 (2001) 996-1007. | MR

[32] E. Hairer, Energy-preserving variant of collocation methods. J. Numer. Anal. Ind. Appl. Math. 5 (2010) 73-84. | MR

[33] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd ed. Springer-Verlag, Berlin (2006). | MR | Zbl

[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. | MR | Zbl

[35] P.E. Hydon and E.L. Mansfield, A variational complex for difference equations. Found. Comput. Math. 4 (2004) 187-217. | MR | Zbl

[36] T. Itoh and K. Abe, Hamiltonian-conserving discrete canonical equations based on variational difference quotients. J. Comput. Phys. 76 (1988) 85-102. | MR | Zbl

[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. | MR | Zbl

[38] C.T. Kelley, Solving nonlinear equations with Newton's method. SIAM, Philadelphia (2003). | MR | Zbl

[39] R.A. Labudde and D. Greenspan, Discrete mechanics-a general treatment. J. Comput. Phys. 15 (1974) 134-167. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[42] L.D. Landau and E.M. Lifshitz, Mechanics, 3rd ed. Butterworth-Heinemann, London (1976). | MR

[43] R.J. Leveque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002). | MR | Zbl

[44] A. Lew, J.E. Marsden, M. Ortiz and M. West, Asynchronous variational integrators. Arch. Ration. Mech. Anal. 167 (2003) 85-146. | MR | Zbl

[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. | MR | Zbl

[46] J.D. Logan, First integrals in the discrete variational calculus. Aequationes Math. 9 (1973) 210-220. | MR | Zbl

[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. | MR | Zbl

[48] J.E. Marsden, G.W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs. Commun. Math. Phys. 199 (1998) 351-395. | MR | Zbl

[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. | MR | Zbl

[50] J.E. Marsden and M. West, Discrete mechanics and variational integrators. Acta Numer. 10 (2001) 357-514. | MR | Zbl

[51] T. Matsuo, High-order schemes for conservative or dissipative systems. J. Comput. Appl. Math. 152 (2003) 305-317. | MR | Zbl

[52] T. Matsuo, New conservative schemes with discrete variational derivatives for nonlinear wave equations. J. Comput. Appl. Math. 203 (2007) 32-56. | MR | Zbl

[53] T. Matsuo, Dissipative/conservative Galerkin method using discrete partial derivative for nonlinear evolution equations. J. Comput. Appl. Math. 218 (2008) 506-521. | MR | Zbl

[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. | MR | Zbl

[55] T. Matsuo, M. Sugihara, D. Furihata and M. Mori, Linearly implicit finite difference schemes derived by the discrete variational method. RIMS Kokyuroku 1145 (2000) 121-129. | MR | Zbl

[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. | MR | Zbl

[57] R.I. Mclachlan, G.R.W. Quispel and N. Robidoux, Geometric integration using discrete gradients. Philos. Trans. Roy. Soc. A 357 (1999) 1021-1046. | MR | Zbl

[58] R.I. Mclachlan and N. Robidoux, Antisymmetry, pseudospectral methods, weighted residual discretizations, and energy conserving partial differential equations, preprint.

[59] K.S. Miller, Linear difference equations, W.A. Benjamin Inc., New York-Amsterdam (1968). | MR | Zbl

[60] P. Olver, Applications of Lie Groups to Differential Equations, 2nd ed. In vol. 107. Graduate Texts in Mathematics. Springer-Verlag, New York (1993). | MR | Zbl

[61] F.A. Potra and W.C. Rheinboldt, On the numerical solution of Euler − Lagrange equations. Mech. Struct. Mach., 19 (1991) 1-18. | MR | Zbl

[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. | MR

[63] G.R.W. Quispel and D.I. Mclaren, A new class of energy-preserving numerical integration methods. J. Phys. A 41 (2008) 045206. | MR | Zbl

[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). | MR | Zbl

[66] L.F. Shampine, Conservation laws and the numerical solution of ODEs. Comput. Math. Appl. B12 (1986) 1287-1296. | MR | Zbl

[67] L.F. Shampine, Conservation laws and the numerical solution of ODEs II. Comput. Math. Appl. 38 (1999) 61-72. | MR | Zbl

[68] M. West, Variational integrators, Ph.D. thesis, California Institute of Technology (2004). | MR

[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. | MR | Zbl

[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. | MR | Zbl

[71] G. Zhong and J.E. Marsden, Lie-Poisson integrators and Lie-Poisson Hamilton-Jacobi theory. Phys. Lett. A 133 (1988) 134-139. | MR