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Matrix-Variation-of-Constants Formula

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Structure-Preserving Algorithms for Oscillatory Differential Equations II

Abstract

This first chapter presents the matrix-variation-of-constants formula, which is fundamental to structure-preserving integrators for multi-frequency and multidimensional oscillatory second-order differential equations in the current volume and the previous volume [23] of our monograph, since the formula makes it possible to incorporate the special structure of the multi-frequency oscillatory problems into the integrators.

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References

  1. Al-Mohy AH, Higham NJ, Relton SD (2015) New algorithms for computing the matrix sine and cosine separately or simultaneously. SIAM J Sci Comput 37:A456–A487

    Article  MathSciNet  MATH  Google Scholar 

  2. Deuflhard P (1979) A study of extrapolation methods based on multistep schemes without parasitic solutions. Z Angew Math Phys 30:177–189

    Article  MathSciNet  MATH  Google Scholar 

  3. García-Archilla B, Sanz-Serna JM, Skeel RD (1998) Long-time-step methods for oscillatory differential equations. SIAM J Sci Comput 20:930–963

    Article  MathSciNet  MATH  Google Scholar 

  4. Hairer E, Lubich C (2000) Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J Numer Anal 38:414–441

    Article  MathSciNet  MATH  Google Scholar 

  5. Hairer E, Lubich C, Wanner G (2006) Geometric numerical integration: structure-preserving algorithms for ordinary differential equations, 2nd edn. Springer, Berlin

    MATH  Google Scholar 

  6. Hairer E, Nørsett SP, Wanner G (1993) Solving ordinary differential equations I: nonstiff problems. Springer, Berlin

    MATH  Google Scholar 

  7. Hochbruck M, Lubich C (1999) A Gautschi-type method for oscillatory second-order differential equations. Numer Math 83:403–426

    Article  MathSciNet  MATH  Google Scholar 

  8. Hochbruck M, Ostermann A (2010) Exponential integrators. Acta Numer 19:209–286

    Article  MathSciNet  MATH  Google Scholar 

  9. Iserles A, Nørsett SP (2005) Efficient quadrature of highly oscillatory integrals using derivatives. Proc R Soc Lond, Ser A, Math Phys Eng Sci 461:1383–1399

    Article  MathSciNet  MATH  Google Scholar 

  10. Jordn DW, Smith P (2007) Nonlinear ordinary differential equations. An introduction for scientists and engineers, 3rd edn. Oxford University Press, Oxford

    Google Scholar 

  11. Li J, Wang B, You X, Wu X (2011) Two-step extended RKN methods for oscillatory systems. Comput Phys Commun 182:2486–2507

    Article  MathSciNet  MATH  Google Scholar 

  12. Liu C, Shi W, Wu X (2014) An efficient high-order explicit scheme for solving Hamiltonian nonlinear wave equations. Appl Math Comput 246:696–710

    Article  MathSciNet  Google Scholar 

  13. Nyström EJ (1925) Ueber die numerische Integration von Differentialgleichungen. Acta Soc Sci Fenn 50:1–54

    Google Scholar 

  14. Wang B, Wu X (2013) A new high precision energy-preserving integrator for system of oscillatory second-order differential equations. Phys Lett A 376:1185–1190

    Article  MathSciNet  MATH  Google Scholar 

  15. Wang B, Wu X, Liu K (2013) A Filon-type asymptotic approach to solving highly oscillatory second-order initial value problems. J Comput Phys 243:210–223

    Article  MathSciNet  Google Scholar 

  16. Wang B, Wu X (2014) Improved Filon-type asymptotic methods for highly oscillatory differential equations with multiple time scales. J Comput Phys 276:62–73

    Article  MathSciNet  Google Scholar 

  17. Wang B, Wu X, Xia J (2013) Error bounds for explicit ERKN integrators for systems of multi-frequency oscillatory second-order differential equations. Appl Numer Math 74:17–34

    Article  MathSciNet  MATH  Google Scholar 

  18. Wang B, Iserles A, Wu X (2014) Arbitrary order trigonometric Fourier collocation methods for multi-frequency oscillatory systems. Found Comput Math. doi:10.1007/s10208-014-9241-9

  19. Wu X (2014) Matrix-variation-of-constants formula with applications. A seminar report of Nanjing University (preprint)

    Google Scholar 

  20. Wu X, Mei L, Liu C (2015) An analytical expression of solutions to nonlinear wave equations in higher dimensions with Robin boundary conditions. J Math Anal Appl 426:1164–1173

    Google Scholar 

  21. Wu X, You X, Shi W, Wang B (2010) ERKN integrators for systems of oscillatory second-order differential equations. Comput Phys Commun 181:1873–1887

    Article  MathSciNet  MATH  Google Scholar 

  22. Wu X, You X, Xia J (2009) Order conditions for ARKN methods solving oscillatory systems. Comput Phys Commun 180:2250–2257

    Article  MathSciNet  MATH  Google Scholar 

  23. Wu X, You X, Wang B (2013) Structure-preserving integrators for oscillatory ordinary differential equations. Springer, Heidelberg (jointly published with Science Press Beijing)

    Google Scholar 

  24. You X, Zhao J, Yang H, Fang Y, Wu X (2014) Order conditions for RKN methods solving general second-order oscillatory systems. Numer Algo 66:147–176

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, China

    Xinyuan Wu

  2. Nanjing University of Finance and Economics, Nanjing, China

    Kai Liu

  3. Nanjing University, Nanjing, China

    Wei Shi

Authors
  1. Xinyuan Wu
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  2. Kai Liu
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  3. Wei Shi
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Correspondence to Xinyuan Wu .

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© 2015 Springer-Verlag Berlin Heidelberg and Science Press, Beijing, China

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Wu, X., Liu, K., Shi, W. (2015). Matrix-Variation-of-Constants Formula. In: Structure-Preserving Algorithms for Oscillatory Differential Equations II. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48156-1_1

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  • DOI: https://doi.org/10.1007/978-3-662-48156-1_1

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-48155-4

  • Online ISBN: 978-3-662-48156-1

  • eBook Packages: EngineeringEngineering (R0)

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