Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-24T10:34:14.787Z Has data issue: false hasContentIssue false
  • English
  • Français

Bespoke finite difference schemes that preserve multiple conservation laws

Part of: Infinite-dimensional Hamiltonian systems Difference equations Difference and functional equations Computational aspects of field theory and polynomials Numerical problems in dynamical systems Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  01 May 2015

Timothy J. Grant
Timothy J. Grant*
Affiliation:
Department of Mathematics, University of Surrey, Guildford GU2 7XH, UK email [email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Conservation laws provide important constraints on the solutions of partial differential equations (PDEs), therefore it is important to preserve them when discretizing such equations. In this paper, a new systematic method for discretizing a PDE, so as to preserve the local form of multiple conservation laws, is presented. The technique, which uses symbolic computation, is applied to the Korteweg–de Vries (KdV) equation to find novel explicit and implicit schemes that have finite difference analogues of its first and second conservation laws and its first and third conservation laws. The resulting schemes are numerically compared with a multisymplectic scheme.

MSC classification

Primary: 39A14: Partial difference equations 37K05: Hamiltonian structures, symmetries, variational principles, conservation laws 65M06: Finite difference methods 65P99: None of the above, but in this section
Secondary: 12Y99: None of the above, but in this section 65M12: Stability and convergence of numerical methods
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 18 , Issue 1 , 2015 , pp. 372 - 403
Copyright
© The Author 2015 

References

Alonso, L. M., ‘On the Noether map’, Lett. Math. Phys. 3 (1979) no. 5, 419–424.Google Scholar
Ascher, U. M. and McLachlan, R. I., ‘Multisymplectic box schemes and the Korteweg–de Vries equation’, Appl. Numer. Math. 48 (2004) no. 3–4, 255–269.CrossRefGoogle Scholar
Ascher, U. M. and McLachlan, R. I., ‘On symplectic and multisymplectic schemes for the KdV equation’, J. Sci. Comput. 25 (2005) no. 1–2, 83–104.Google Scholar
Bridges, T. J. and Reich, S., ‘Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity’, Phys. Lett. A 284 (2001) no. 4–5, 184–193.Google Scholar
Bridges, T. J. and Reich, S., ‘Numerical methods for Hamiltonian PDEs’, J. Phys. A 39 (2006) no. 19, 5287–5320.Google Scholar
Buchberger, B. and Kauers, M., ‘Groebner basis’, Scholarpedia 5 (2010) no. 10, 7763.Google Scholar
Buchberger, B. and Kauers, M., ‘Buchberger’s algorithm’, Scholarpedia 6 (2011) no. 10, 7764.Google Scholar
Budd, C. J. and Piggott, M. D., ‘Geometric integration and its applications’, Handbook of numerical analysis , Handbook of Numerical Analysis XI (North-Holland, Amsterdam, 2003) 35–139.Google Scholar
Cox, D., Little, J. and O’Shea, D., ‘An introduction to computational algebraic geometry and commutative algebra’, Ideals, varieties, and algorithms , 3rd edn, Undergraduate Texts in Mathematics (Springer, New York, 2007).CrossRefGoogle Scholar
Drazin, P. G. and Johnson, R. S., Solitons: an introduction , Cambridge Texts in Applied Mathematics (Cambridge University Press, Cambridge, 1989).Google Scholar
Duzhin, S. V. and Tsujishita, T., ‘Conservation laws of the BBM equation’, J. Phys. A 17 (1984) no. 16, 3267–3276.CrossRefGoogle Scholar
de Frutos, J. and Sanz-Serna, J. M., ‘Accuracy and conservation properties in numerical integration: the case of the Korteweg–de Vries equation’, Numer. Math. 75 (1997) no. 4, 421–445.Google Scholar
Furihata, D., ‘Finite difference schemes for ∂u∕∂t = (∂∕∂x)𝛼𝛿G∕𝛿u that inherit energy conservation or dissipation property’, J. Comput. Phys. 156 (1999) no. 1, 181–205.Google Scholar
Grant, T. J., ‘Characteristics of conservation laws for finite difference equations’, PhD Thesis, Department of Mathematics, University of Surrey, 2011.Google Scholar
Grant, T. J. and Hydon, P. E., ‘Characteristics of conservation laws for difference equations’, Found. Comput. Math. 13 (2013) no. 4, 667–692.CrossRefGoogle Scholar
Hairer, E., Lubich, C. and Wanner, G., ‘Structure-preserving algorithms for ordinary differential equations’, Geometric numerical integration , 2nd edn, Springer Series in Computational Mathematics 31 (Springer, Berlin, 2006).Google Scholar
Hydon, P. E., ‘Conservation laws of partial difference equations with two independent variables’, J. Phys. A 34 (2001) no. 48, 10347–10355.Google Scholar
Hydon, P. E. and Mansfield, E. L., ‘A variational complex for difference equations’, Found. Comput. Math. 4 (2004) no. 2, 187–217.CrossRefGoogle Scholar
Koide, S. and Furihata, D., ‘Nonlinear and linear conservative finite difference schemes for regularized long wave equation’, Jpn. J. Ind. Appl. Math. 26 (2009) no. 1, 15–40.Google Scholar
Kuperschmidt, B. A., ‘Discrete Lax equations and differential-difference calculus’, Astérisque (1985) no. 123, 212.Google Scholar
Leimkuhler, B. and Reich, S., Simulating Hamiltonian dynamics , Cambridge Monographs on Applied and Computational Mathematics 14 (Cambridge University Press, Cambridge, 2004).Google Scholar
Mansfield, E., ‘Differential Groebner bases’, PhD Thesis, 1991.Google Scholar
McLachlan, R. I., ‘Spatial discretization of partial differential equations with integrals’, IMA J. Numer. Anal. 23 (2003) no. 4, 645–664.Google Scholar
McLachlan, R. and Quispel, R., ‘Six lectures on the geometric integration of ODEs’, Foundations of computational mathematics (Oxford, 1999) , London Mathematical Society Lecture Note Series 284 (Cambridge University Press, Cambridge, 2001) 155–210.Google Scholar
Noether, E., ‘Invariant variation problems’, Transport Theory Statist. Phys. 1 (1971) no. 3, 186–207; Translated from the German (Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1918, 235–257).Google Scholar
Olver, P. J., ‘Euler operators and conservation laws of the BBM equation’, Math. Proc. Cambridge Philos. Soc. 85 (1979) no. 1, 143–160.Google Scholar
Olver, P. J., Applications of Lie groups to differential equations , 2nd edn, Graduate Texts in Mathematics 107 (Springer, New York, 1993).Google Scholar
Sanz-Serna, J. M., ‘An explicit finite-difference scheme with exact conservation properties’, J. Comput. Phys. 47 (1982) 199–210.CrossRefGoogle Scholar
Vliegenthart, A. C., ‘On finite-difference methods for the Korteweg–de Vries equation’, J. Engrg. Math. 5 (1971) 137–155.Google Scholar
Zabusky, N. J. and Kruskal, M. D., ‘Interaction of “solitons” in a collisionless plasma and the recurrence of initial states’, Phys. Rev. Lett. 15 (1965) no. 6, 240–243.Google Scholar
Zhong, G. and Marsden, J. E., ‘Lie–Poisson Hamilton–Jacobi theory and Lie–Poisson integrators’, Phys. Lett. A 133 (1988) no. 3, 134–139.Google Scholar