Abstract
A new exponentially fitted version of the discrete variational derivative method for the efficient solution of oscillatory complex Hamiltonian partial differential equations is proposed. When applied to the nonlinear Schrödinger equation, this scheme has discrete conservation laws of charge and energy. The new method is compared with other conservative schemes from the literature on a benchmark problem whose solution is an oscillatory breather wave.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Akhmediev, N.N., Eleonskiĭ, V.M., Kulagin, N.E.: First-order exact solutions of the nonlinear Schrödinger equation. Teoret. Mat. Fiz. 72(2), 183–196 (1987). https://doi.org/10.1103/PhysRevA.47.3213
Ascher, U.M., McLachlan, R.I.: On symplectic and multisymplectic schemes for the KdV equation. J. Sci. Comput. 25(1–2), 83–104 (2005). https://doi.org/10.1007/s10915-004-4634-6
Bodurov, T.: Complex Hamiltonian evolution equations and field theory. J. Math. Phys. 39(11), 5700–5715 (1998). https://doi.org/10.1063/1.532587
Bodurov, T.: Derivation of the nonlinear Schrödinger equation from first principles. Ann. Fond. Louis de Broglie 30(3–4), 343–352 (2005)
Bridges, T.J., Reich, S.: Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity. Phys. Lett. A 284(4–5), 184–193 (2001). https://doi.org/10.1016/S0375-9601(01)00294-8
Bridges, T.J., Reich, S.: Numerical methods for Hamiltonian PDEs. J. Phys. A 39(19), 5287–5320 (2006). https://doi.org/10.1088/0305-4470/39/19/S02
Brugnano, L., Iavernaro, F.: Line integral methods for conservative problems. CRC Press, Boca Raton, FL (2016)
Brugnano, L., Frasca-Caccia, G., Iavernaro, F.: Line integral solution of Hamiltonian PDEs. Mathematics, 7(3), (2019a). https://doi.org/10.3390/math7030275
Brugnano, L., Iavernaro, F., Montijano, J.I., Rández, L.: Spectrally accurate space-time solution of Hamiltonian PDEs. Numer. Algorithms 81(4), 1183–1202 (2019). https://doi.org/10.1007/s11075-018-0586-z
Burrage, K., Cardone, A., D’Ambrosio, R., Paternoster, B.: Numerical solution of time fractional diffusion systems. Appl. Numer. Math. 116, 82–94 (2017). https://doi.org/10.1016/j.apnum.2017.02.004
Cardone, A., Ixaru, L.Gr., Paternoster, B.: Exponential fitting direct quadrature methods for Volterra integral equations. Numer. Algorithms 55(4), 467–480 (2010). https://doi.org/10.1007/s11075-010-9365-1
Cardone, A., Ixaru, L.Gr., Paternoster, B., Santomauro, G.: Ef-Gaussian direct quadrature methods for Volterra integral equations with periodic solution. Math. Comput. Simulation 110, 125–143 (2015). https://doi.org/10.1016/j.matcom.2013.10.005
Cardone, A., D’Ambrosio, R., Paternoster, B.: Exponentially fitted IMEX methods for advection-diffusion problems. J. Comput. Appl. Math. 316, 100–108 (2017). https://doi.org/10.1016/j.cam.2016.08.025
Celledoni, E., Grimm, V., McLachlan, R.I., McLaren, D.I., O’Neale, D., Owren, B., Quispel, G.R.W.: Preserving energy resp. dissipation in numerical PDEs using the “Average Vector Field" method. J. Comput. Phys. 231(20), 6770–6789 (2012). https://doi.org/10.1016/j.jcp.2012.06.022
Conte, D., Frasca-Caccia, G.: Exponentially fitted methods that preserve conservation laws. Commun. Nonlinear Sci. Numer. Simul. 109, 106334 (2022). https://doi.org/10.1016/j.cnsns.2022.106334
Conte, D., Paternoster, B.: Modified Gauss-Laguerre exponential fitting based formulae. J. Sci. Comput. 69(1), 227–243 (2016). https://doi.org/10.1007/s10915-016-0190-0
Conte, D., Ixaru, L.Gr., Paternoster, B., Santomauro, G.: Exponentially-fitted Gauss-Laguerre quadrature rule for integrals over an unbounded interval. J. Comput. Appl. Math. 255, 725–736 (2014). https://doi.org/10.1016/j.cam.2013.06.040
Conte, D., D’Ambrosio, R., Moccaldi, M., Paternoster, B.: Adapted explicit two-step peer methods. J. Numer. Math. 27(2), 69–83 (2019). https://doi.org/10.1515/jnma-2017-0102
Conte, D., Mohammadi, F., Moradi, L., Paternoster, B.: Exponentially fitted two-step peer methods for oscillatory problems. Comput. Appl. Math., 39(3):Paper No. 174, 19, 2020. https://doi.org/10.1007/s40314-020-01202-x
Conte, D., Pagano, G., Paternoster, B.: Nonstandard finite differences numerical methods for a vegetation reaction-diffusion model. J. Comput. Appl. Math., 419, 2023. https://doi.org/10.1016/j.cam.2022.114790
Dahlby, M., Owren, B.: A general framework for deriving integral preserving numerical methods for PDEs. SIAM J. Sci. Comput. 33(5), 2318–2340 (2011). https://doi.org/10.1137/100810174
D’Ambrosio, R., Paternoster, B.: Numerical solution of a diffusion problem by exponentially fitted finite difference methods. SpringerPlus 3(1), 425 (2014). https://doi.org/10.1186/2193-1801-3-425
D’Ambrosio, R., Esposito, E., Paternoster, B.: Parameter estimation in exponentially fitted hybrid methods for second order differential problems. J. Math. Chem. 50(1), 155–168 (2012). https://doi.org/10.1007/s10910-011-9903-7
De Frutos, J., Sanz-Serna, J.M.: Accuracy and conservation properties in numerical integration: the case of the Korteweg-de Vries equation. Numer. Math. 75(4), 421–445 (1997). https://doi.org/10.1007/s002110050247
De Meyer, H., Vanthournout, J., Vanden Berghe, G.: On a new type of mixed interpolation. J. Comput. Appl. Math. 30(1), 55–69 (1990). https://doi.org/10.1016/0377-0427(90)90005-K
Diele, F., Marangi, C.: Positive symplectic integrators for predator-prey dynamics. Discrete Continuous Dyn. Syst. Ser. B, 23(7), (2018). https://doi.org/10.3934/dcdsb.2017185
Durán, A., Sanz-Serna, J.M.: The numerical integration of relative equilibrium solutions. Geometric theory. Nonlinearity 11(6), 1547–1567 (1998). https://doi.org/10.1088/0951-7715/11/6/008
Durán, A., Sanz-Serna, J.M.: The numerical integration of relative equilibrium solutions. The nonlinear Schrödinger equation. IMA J. Numer. Anal. 20(2), 235–261 (2000). https://doi.org/10.1093/imanum/20.2.235
Evans, G.A., Webster, J.R.: A high order, progressive method for the evaluation of irregular oscillatory integrals. Appl. Numer. Math. 23(2), 205–218 (1997). https://doi.org/10.1016/S0168-9274(96)00058-X
Frasca-Caccia, G., Hydon, P.E.: Locally conservative finite difference schemes for the modified KdV equation. J. Comput. Dyn. 6(2), 307–323 (2019). https://doi.org/10.3934/jcd.2019015
Frasca-Caccia, G., Hydon, P.E.: Simple bespoke preservation of two conservation laws. IMA J. Numer. Anal. 40(2), 1294–1329 (2020). https://doi.org/10.1093/imanum/dry087
Frasca-Caccia, G., Hydon, P.E.: Numerical preservation of multiple local conservation laws. Appl. Math. Comput. 403, 126203 (2021). https://doi.org/10.1016/j.amc.2021.126203
Frasca-Caccia, G., Hydon, P.E.: A new technique for preserving conservation laws. Found. Comput. Math. 22, 477–506 (2022). https://doi.org/10.1007/s10208-021-09511-1
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). https://doi.org/10.1006/jcph.1999.6377
Furihata, D., Matsuo, T.: Discrete variational derivative method. CRC Press, Boca Raton, FL (2011)
Gonzalez, O.: Time integration and discrete Hamiltonian systems. J. Nonlinear Sci. 6(5), 449–467 (1996). https://doi.org/10.1007/s003329900018
Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration, 2nd edn. Structure preserving algorithms for ordinary differential equations. Springer, Berlin (2006)
Hollevoet, D., Van Daele, M., Vanden Berghe, G.: Exponentially fitted methods applied to fourth-order boundary value problems. J. Comput. Appl. Math. 235(18), 5380–5393 (2011). https://doi.org/10.1016/j.cam.2011.05.049
Ixaru, L.Gr., Vanden Berghe, G.: Exponential fitting. Kluwer Academic Publishers, Dordrecht, (2004)
Leimkuhler, B., Reich, S.: Simulating Hamiltonian dynamics. Cambridge University Press, Cambridge (2004)
Mansfield, E.L., Rojo-Echeburua, A., Hydon, P.E., Peng, L.: Moving frames and Noether’s finite difference conservation laws I. Trans. Math. Appl. 3(1), 1–47 (2019). https://doi.org/10.1093/imatrm/tnz004
Matsuo, T.: New conservative schemes with discrete variational derivatives for nonlinear wave equations. J. Comput. Appl. Math. 203(1), 32–56 (2007). https://doi.org/10.1016/j.cam.2006.03.009
Matsuo, T., Furihata, D.: Dissipative or conservative finite-difference schemes for complex-valued nonlinear partial differential equations. J. Comput. Phys. 171(2), 425–447 (2001). https://doi.org/10.1006/jcph.2001.6775
McLachlan, R.I., Quispel, G.R.W.: Discrete gradient methods have an energy conservation law. Discrete Contin. Dyn. Syst. 34(3), 1099–1104 (2014). https://doi.org/10.3934/dcds.2014.34.1099
McLachlan, R.I., Stern, A.: Functional equivariance and conservation laws in numerical integration. Found. Comput. Math. (2022). https://doi.org/10.1007/s10208-022-09590-8
McLachlan, R.I., Quispel, G.R.W., Robidoux, N.: Geometric integration using discrete gradients R. Soc. Lond Philos. Trans. Ser. A Math. Phys. Eng. Sci. 357(1754), 1021–1045 (1999). https://doi.org/10.1098/rsta.1999.0363
Mickens, R.E.: Nonstandard finite difference models of differential equations. World Scientific, Singapore (1994)
Miyatake, Y.: An energy-preserving exponentially-fitted continuous stage Runge-Kutta method for Hamiltonian systems. BIT 54(3), 777–799 (2014). https://doi.org/10.1007/s10543-014-0474-4
Paternoster, B.: Present state-of-the-art in exponential fitting. A contribution dedicated to Liviu Ixaru on his 70th birthday. Comput. Phys. Commun. 183(12), 2499–2512 (2012). https://doi.org/10.1016/j.cpc.2012.06.013
Qin, T., Hua, Y., Zhang, M.: A class of adaptive exponentially fitted rosenbrock methods with variable coefficients for symmetric systems. Symmetry 14(8), 1708 (2022). https://doi.org/10.3390/sym14081708
Quispel, G.R.W., McLaren, D.I.: A new class of energy-preserving numerical integration methods. J. Phys. A 41(4), 045206 (2008). https://doi.org/10.1088/1751-8113/41/4/045206
Rebelo, R., Valiquette, F.: Symmetry preserving numerical schemes for partial differential equations and their numerical tests. J. Differ. Equ. Appl. 19(5), 738–757 (2013). https://doi.org/10.1080/10236198.2012.685470
Simos, T.E.: An exponentially-fitted Runge-Kutta method for the numerical integration of initial-value problems with periodic or oscillating solutions. Comput. Phys. Comm. 115(1), 1–8 (1998). https://doi.org/10.1016/S0010-4655(98)00088-5
Van Daele, M., Vanden Berghe, G.: Geometric numerical integration by means of exponentially-fitted methods. Appl. Numer. Math. 57(4), 415–435 (2007). https://doi.org/10.1016/j.apnum.2006.06.001
Vanden Berghe, G., Ixaru, L.Gr., De Meyer, H.: Frequency determination and step-length control for exponentially-fitted Runge-Kutta methods. J. Comput. Appl. Math. 132(1), 95–105 (2001). https://doi.org/10.1016/S0377-0427(00)00602-6
Wan, A.T., 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). https://doi.org/10.1137/140997944
Yin, X., Liu, Y., Zhang, J., Shen, Y., Yan, L.: Exponentially fitted multisymplectic scheme for conservative maxwell equations with oscillary solutions. PLoS ONE 16(8), e0256108 (2021). https://doi.org/10.1371/journal.pone.0256108
Acknowledgements
This work is supported by the GNCS-INDAM project and by the PRIN2017-MIUR project. The authors are members of the INdAM research group GNCS.
Funding
Open access funding provided by Universitá degli Studi di Salerno within the CRUI-CARE Agreement.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no competing interests.
Additional information
Communicated by: Akil Narayan
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Conte, D., Frasca-Caccia, G. Exponentially fitted methods with a local energy conservation law. Adv Comput Math 49, 49 (2023). https://doi.org/10.1007/s10444-023-10049-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10444-023-10049-9