Linear gradient structures and discrete gradient methods for conservative/dissipative differential-algebraic equations

  • Shun SatoEmail author


In this paper, the use of discrete gradients is considered for differential-algebraic equations (DAEs) with a conservation/dissipation law. As one of the most popular numerical methods for conservative/dissipative ordinary differential equations, the framework of the discrete gradient method has been intensively developed over recent decades. Although discrete gradients have been applied to several specific DAEs, no unified framework has yet been constructed. In this paper, the author moves toward the establishment of such a framework, and introduces concepts including an appropriate linear gradient structure for DAEs. Then, it is revealed that the simple use of discrete gradients does not imply the discrete conservation/dissipation laws. Fortunately, however, for the case of index-1 DAEs, an appropriate reformulation and a new discrete gradient enable us to successfully construct a novel scheme, which satisfies both of the discrete conservation/dissipation law and the constraint. This first attempt may provide an indispensable basis for constructing a unified framework of discrete gradient methods for DAEs.


Discrete gradient method Differential-algebraic equations Linear gradient form Conservation law Dissipation law 

Mathematics Subject Classification




The author is grateful to Kensuke Aishima and Takayasu Matsuo for valuable comments. The author thanks the anonymous reviewers for many helpful comments.


  1. 1.
    Ascher, U.M., Petzold, L.R.: Computer Methods for Ordinary Differential Equations and Differential-algebraic Equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1998). CrossRefzbMATHGoogle Scholar
  2. 2.
    Bajić, V.B.: Lyapunov function candidates for semistate systems. Int. J. Control 46(6), 2171–2181 (1987). MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ben-Israel, A., Greville, T.N.E.: Generalized Inverses. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 15, 2nd edn. Springer, New York (2003)Google Scholar
  4. 4.
    Betsch, P.: Energy-consistent numerical integration of mechanical systems with mixed holonomic and nonholonomic constraints. Comput. Methods Appl. Mech. Eng. 195(50–51), 7020–7035 (2006). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bloch, A.M.: Nonholonomic Mechanics and Control. Interdisciplinary Applied Mathematics, vol. 24, 2nd edn. Springer, New York (2015). CrossRefGoogle Scholar
  6. 6.
    Burger, M., Gerdts, M.: A survey on numerical methods for the simulation of initial value problems with sDAEs. In: Ilchmann, A., Reis, T. (eds.) Surveys in Differential-Algebraic Equations, vol. IV, pp. 221–300. Springer, Cham (2017)CrossRefGoogle Scholar
  7. 7.
    Celledoni, E., Eidnes, S., Owren, B., Ringholm, T.: Dissipative numerical schemes on Riemannian manifolds with applications to gradient flows. SIAM J. Sci. Comput. 40(6), A3789–A3806 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Celledoni, E., Eidnes, S., Owren, B., Ringholm, T.: Energy preserving methods on Riemannian manifolds. eprints. arXiv:1805.07578 (2018)
  9. 9.
    Celledoni, E., Grimm, V., McLachlan, R., McLaren, D., O’Neale, D., Owren, B., Quispel, G.: Preserving energy resp. dissipation in numerical PDEs using the “average vector field” method. J. Comput. Phys. 231, 6770–6789 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Celledoni, E., Owren, B.: Preserving first integrals with symmetric Lie group methods. Discrete Contin. Dyn. Syst. 34(3), 977–990 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Furihata, D.: Finite difference schemes for \(\partial u/\partial t=(\partial /\partial x)^\alpha \delta G/\delta u\) that inherit energy conservation or dissipation property. J. Comput. Phys. 156(1), 181–205 (1999). MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Furihata, D., Matsuo, T.: Discrete Variational Derivative Method—A Structure-Preserving Numerical Method for Partial Differential Equations. CRC Press, Boca Raton (2011)zbMATHGoogle Scholar
  13. 13.
    Furihata, D., Mori, M.: General derivation of finite difference schemes by means of a discrete variation. Trans. Jpn. Soc. Ind. Appl. 8(3), 317–340 (1998) (in Japanese)Google Scholar
  14. 14.
    Furihata, D., Sato, S., Matsuo, T.: A novel discrete variational derivative method using “average-difference methods”. JSIAM Lett. 8, 81–84 (2016). 10.14495/jsiaml.8.81MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gonzalez, O.: Time integration and discrete Hamiltonian systems. J. Nonlinear Sci. 6, 449–467 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gonzalez, O.: Mechanical systems subject to holonomic constraints: differential-algebraic formulations and conservative integration. Phys. D 132(1–2), 165–174 (1999). MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, vol. 31. Springer, Heidelberg (2010)zbMATHGoogle Scholar
  18. 18.
    Ishikawa, A., Yaguchi, T.: Application of the variational principle to deriving energy-preserving schemes for the Hamilton equation. JSIAM Lett. 8, 53–56 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Itoh, T., Abe, K.: Hamiltonian-conserving discrete canonical equations based on variational difference quotients. J. Comput. Phys. 76(1), 85–102 (1988). MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kojima, H.: Invariants preserving schemes based on explicit Runge–Kutta methods. BIT 56(4), 1317–1337 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Li, M., Yin, Z.: Blow-up phenomena and travelling wave solutions to the periodic integrable dispersive Hunter–Saxton equation. Discrete Contin. Dyn. Syst. 37(12), 6471–6485 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Liberzon, D., Trenn, S.: Switched nonlinear differential algebraic equations: solution theory, Lyapunov functions, and stability. Autom. J. IFAC 48(5), 954–963 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    McLachlan, R.I., Quispel, G.R.W., Robidoux, N.: Geometric integration using discrete gradients. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 357, 1021–1045 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Miyatake, Y., Cohen, D., Furihata, D., Matsuo, T.: Geometric numerical integrators for Hunter–Saxton-like equations. Jpn. J. Ind. Appl. Math. 34(2), 441–472 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Miyatake, Y., Yaguchi, T., Matsuo, T.: Numerical integration of the Ostrovsky equation based on its geometric structures. J. Comput. Phys. 231(14), 4542–4559 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Olver, P.J.: Equivalence, Invariants, and Symmetry. Cambridge University Press, Cambridge (1995). CrossRefzbMATHGoogle Scholar
  27. 27.
    Quispel, G.R.W., Capel, H.W.: Solving ODE’s numerically while preserving a first integral. Phys. Lett. A 218, 223–228 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Quispel, G.R.W., McLaren, D.I.: A new class of energy-preserving numerical integration methods. J. Phys. A. Math. Theor. 41, 045,206 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Quispel, G.R.W., Turner, G.S.: Discrete gradient methods for solving ODE’s numerically while preserving a first integral. J. Phys. A 29, L341–349 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Reich, S.: On a geometrical interpretation of differential-algebraic equations. Circuits Syst. Signal Process. 9(4), 367–382 (1990). MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Robinson, J.C.: Infinite-Dimensional Dynamical Systems. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  32. 32.
    Sato, S.: Stability and convergence of a conservative finite difference scheme for the modified Hunter-Saxton equation. BIT 59(1), 213–241 (2019). MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Sato, S., Matsuo, T.: On spatial discretization of evolutionary differential equations on the periodic domain with a mixed derivative. J. Comput. Appl. Math. 358, 221–240 (2019)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Sato, S., Matsuo, T., Suzuki, H., Furihata, D.: A Lyapunov-type theorem for dissipative numerical integrators with adaptive time-stepping. SIAM J. Numer. Anal. 53(6), 2505–2518 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Uhlar, S., Betsch, P.: On the derivation of energy consistent time stepping schemes for friction afflicted multibody systems. Comput. Struct. 88(11), 737–754 (2010). CrossRefGoogle Scholar
  36. 36.
    Wan, A.T.S., Bihlo, A., Nave, J.C.: The multiplier method to construct conservative finite difference schemes for ordinary and partial differential equations. SIAM J. Numer. Anal. 54(1), 86–119 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Wan, A.T.S., Bihlo, A., Nave, J.C.: Conservative methods for dynamical systems. SIAM J. Numer. Anal. 55(5), 2255–2285 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Wan, A.T.S., Nave, J.C.: On the arbitrarily long-term stability of conservative methods. eprints. arXiv:1607.06160 (2016)

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematical Informatics, Graduate School of Information Science and TechnologyThe University of TokyoBunkyo-kuJapan

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