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Second Order Conformal Symplectic Schemes for Damped Hamiltonian Systems

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Abstract

Numerical methods for solving linearly damped Hamiltonian systems are constructed using the popular Störmer–Verlet and implicit midpoint methods. Each method is shown to preserve dissipation of symplecticity and dissipation of angular momentum of an N-body system with pairwise distance dependent interactions. Necessary and sufficient conditions for second order accuracy are derived. Analysis for linear equations gives explicit relationships between the damping parameter and the step size to reveal when the methods are most advantageous; essentially, the damping rate of the numerical solution is exactly preserved under these conditions. The methods are applied to several model problems, both ODEs and PDEs. Additional structure preservation is discovered for the discretized PDEs, in one case dissipation in total linear momentum and in another dissipation in mass are preserved by the methods. The numerical results, along with comparisons to standard Runge–Kutta methods and another structure-preserving method, demonstrate the usefulness and strengths of the methods.

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References

  1. Armero, F., Romero, I.: On the formulation of high-frequency dissipative time-stepping algorithms for nonlinear dynamics. I. Low-order methods for two model problems and nonlinear elastodynamics. Comput. Methods Appl. Mech. Eng. 190, 2603–2649 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Armero, F., Romero, I.: On the formulation of high-frequency dissipative time-stepping algorithms for nonlinear dynamics. II. Second-order methods. Comput. Methods Appl. Mech. Eng. 190, 6783–6824 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ascher, U.M., McLachlan, R.I.: Multi-symplectic box schemes and the Korteweg–de Vries equation. Appl. Numer. Math. 48, 255–269 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Eden, A., Milani, A., Nicolaenko, B.: Finite dimensional exponential attractors for semilinear wave equations with damping. J. Math. Anal. Appl. 169, 408–419 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Furihata, D.: Finite difference schemes for \(\frac{\partial u}{\partial t} =\left(\frac{\partial }{\partial x}\right)^{\alpha } \frac{\delta G}{\delta u}\) that inherit energy conservation or dissipation property. J. Comput. Phys. 156, 181–205 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kane, C., Marsden, J.E., Ortiz, M., West, M.: Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems. Int. J. Numer. Meth. Eng. 49, 1295–1325 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kong, X., Wu, H., Mei, F.: Structure-preserving algorithms for Birkhoffian systems. J. Geom. Phys. 62, 1157–1166 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Matsuo, T., Furihata, D.: Dissipative or conservative finite-difference schemes for complex-valued nonlinear partial differential equations. J. Comput. Phys. 171, 425–447 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. McLachlan, R.I., Perlmutter, M.: Conformal Hamiltonian systems. J. Geom. Phys. 39, 276–300 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. McLachlan, R.I., Quispel, G.R.W.: What kinds of dynamics are there? Lie pseudogroups, dynamical systems and geometric integration. Nonlinearity 14, 1689–1705 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. McLachlan, R.I., Quispel, G.R.W.: Splitting methods. Acta Numer. 11, 341–434 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Mickens, R.E.: Nonstandard Finite Difference Models of Differential Equations. World Scientific, Singapore (1994)

    MATH  Google Scholar 

  13. Modin, K., Söderlind, G.: Geometric integration of Hamiltonian systems perturbed by Rayleigh damping. BIT Numer. Math. 51, 977–1007 (2011)

    Article  MATH  Google Scholar 

  14. Moore, B.E.: Conformal multi-symplectic integration methods for forced-damped semi-linear wave equations. Math. Comput. Simul. 80, 20–28 (2009)

    Article  MATH  Google Scholar 

  15. Moore, B.E., Noreña, L., Schober, C.: Conformal conservation laws and geometric integration for damped Hamiltonian PDEs. J. Comput. Phys. 232, 214–233 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Su, H., Qin, M., Wang, Y., Scherer, R.: Multi-symplectic Birkhoffian structure for PDEs with dissipation terms. Phys. Lett. A 374, 2410–2416 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sun, Y., Shang, Z.: Structure-preserving algorithms for Birkhoffian systems. Phys. Lett. A 336, 358–369 (2005)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Central Florida, Orlando, Florida, USA

    Ashish Bhatt, Dwayne Floyd & Brian E. Moore

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  1. Ashish Bhatt
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  2. Dwayne Floyd
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  3. Brian E. Moore
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Correspondence to Brian E. Moore.

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Bhatt, A., Floyd, D. & Moore, B.E. Second Order Conformal Symplectic Schemes for Damped Hamiltonian Systems. J Sci Comput 66, 1234–1259 (2016). https://doi.org/10.1007/s10915-015-0062-z

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  • DOI: https://doi.org/10.1007/s10915-015-0062-z

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