A Lyapunov and Sacker–Sell spectral stability theory for one-step methods

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Abstract

Approximation theory for Lyapunov and Sacker–Sell spectra based upon QR techniques is used to analyze the stability of a one-step method solving a time-dependent (nonautonomous) linear ordinary differential equation (ODE) initial value problem in terms of the local error. Integral separation is used to characterize the conditioning of stability spectra calculations. The stability of the numerical solution by a one-step method of a nonautonomous linear ODE using real-valued, scalar, nonautonomous linear test equations is justified. This analysis is used to approximate exponential growth/decay rates on finite and infinite time intervals and establish global error bounds for one-step methods approximating uniformly, exponentially stable trajectories of nonautonomous and nonlinear ODEs. A time-dependent stiffness indicator and a one-step method that switches between explicit and implicit Runge–Kutta methods based upon time-dependent stiffness are developed based upon the theoretical results.

Keywords

One-step methods Stiffness Lyapunov exponents Sacker–Sell spectrum Nonautonomous differential equations 

Mathematics Subject Classification

65L04 65L05 65P40 34D08 34D09 

References

  1. 1.
    Adrianova, L.: Introduction to Linear Systems of Differential Equations, vol. 146. AMS, Providence (1995)MATHGoogle Scholar
  2. 2.
    Arimoto, S., Nagumo, J., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. Inst. Radio Eng. 50, 2060–2070 (1964)Google Scholar
  3. 3.
    Aulbach, B., Wanner, T.: Invariant foliations for Carathéodory type differential equations in Banach spaces. In: Martynyuk, A. (ed.), Advances in Stability Theory at the End of the 20th Century, Stability Control Theory Methods Appl. 13. Taylor and Francis, New York (1999)Google Scholar
  4. 4.
    Badawy, M., Van Vleck, E.S.: Perturbation theory for the approximation of stability spectra by QR methods for sequences of linear operators on a Hilbert space. Linear Algebra Appl. 437(1), 37–59 (2012)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Beyn, W.-J.: On invariant close curves for one-step methods. Numer. Math. 51, 103–122 (1987)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bogacki, P., Shampine, L.: A 3(2) pair of Runge–Kutta formulas. Appl. Math. Lett. 2(4), 321–325 (1989)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Breda, D., Van Vleck, E.S.: Approximating Lyapunov exponents and Sacker–Sell spectrum for retarded functional differential equations. Numer. Math. 126(2), 225–257 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Burrage, K., Butcher, J.C.: Stability criteria for implicit Runge–Kutta methods. SIAM J. Numer. Anal. 16(1), 46–57 (1979)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Butcher, J.C.: A stability property of implicit Runge–Kutta methods. BIT Numer. Math. 27, 358–361 (1975)CrossRefMATHGoogle Scholar
  10. 10.
    Butcher, J.C.: The equivalence of algebraic stability and AN-stability. BIT Numer. Math. 27(2), 510–533 (1987)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Calvo, M., Jay, L., Söderlind, G.: Stiffness 1952–2012: sixty years in search of a definition. BIT Numer. Math. 55(2), 531–558 (2015)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Cameron, F., Palmroth, M., Piché, R.: Quasi stage order conditions for SDIRK methods. Appl. Numer. Math. 42(1–3), 61–75 (2002)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Carpenter, M.H., Kennedy, C.A.: Additive Runge–Kutta schemes for convection–diffusion–reaction equations. Appl. Numer. Math. 44, 139–181 (2003)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Coppel, W.A.: Lecture Notes in Mathematics # 629: Dichotomies in Stability Theory, vol. 629. Springer, Berlin (1978)Google Scholar
  15. 15.
    Dahlquist, G.: Convergence and stability in the numerical integration of ordinary differential equations. Math. Scan. 4, 33–53 (1956)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Dahlquist, G.: Stability and error bounds in the numerical integration of ordinary differential equations. Trans. Royal Inst. Technol., Stockholm, Sweden, Nr. 130: 87 (1959)Google Scholar
  17. 17.
    Dahlquist, G.: A special stability problem for linear multistep methods. BIT Numer. Math. 3, 27–43 (1963)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Dieci, L., Van Vleck, E.S.: Unitary integrators and applications to continuous orthonormalization techniques. SIAM J. Numer. Anal. 310(1), 261–281 (1994)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Dieci, L., Van Vleck, E.S.: Computation of a few Lyapunov exponents for continuous and discrete dynamical systems. Appl. Numer. Math. 17(3), 275–291 (1995)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Dieci, L., Van Vleck, E.S.: Computation of orthonormal factors for fundamental matrix solutions. Numer. Math. 83(4), 599–620 (1999)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Dieci, L., Van Vleck, E.S.: Lyapunov and other spectra: a survey. Collected Lectures on the Preservation of Stability under Discretization, A Volume Published by SIAM, pp. 197–218 (2002)Google Scholar
  22. 22.
    Dieci, L., Van Vleck, E.S.: Lyapunov spectral intervals: theory and computation. SIAM J. Numer. Anal. 40(2), 516–542 (2003)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Dieci, L., Van Vleck, E.S.: On the error in computing Lyapunov exponents by QR methods. Numer. Math. 101(4), 619–642 (2005)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Dieci, L., Van Vleck, E.S.: Lyapunov and Sacker–Sell spectral intervals. J. Dyn. Differ. Equ. 19(2), 265–293 (2007)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Dieci, L., Russell, R., Van Vleck, E.S.: On the computation of Lyapunov exponents for continuous dynamical systems. SIAM J. Numer. Anal. 34(1), 402–423 (1997)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Dieci, L., Elia, C., Van Vleck, E.S.: Detecting exponential dichotomy on the real line: SVD and QR algorithms. BIT Numer. Math. 248, 555–579 (2011)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Eirola, T.: Invariant curves of one-step methods. BIT Numer. Math. 28(1), 113–122 (1988)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Fitzhugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1(6), 445–466 (1961)CrossRefGoogle Scholar
  29. 29.
    Kloeden, P., Lorenz, J.: Stable attracting sets in dynamical systems and in their one-step discretizations. SIAM J. Numer. Anal. 23(5), 986–995 (1986)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Kreiss, H.-O.: Difference methods for stiff ordinary differential equations. SIAM J. Numer. Anal. 15(1), 21–58 (1978)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Krupa, M., Sandstede, B., Szmolyan, P.: Fast and slow waves in the Fitzhugh–Nagumo equation. J. Differ. Equ. 133(1), 49–97 (1997)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Leonov, G.A., Kuznetsov, N.V.: Time-varying linearization and the Perron effects. Int. J. Bifur. Chaos Appl. Sci. Eng. 37(4), 1079–1107 (2007)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Lyapunov, A.: Problém géneral de la stabilité du mouvement. Int. J. Control 53(3), 531–773 (1992)CrossRefGoogle Scholar
  34. 34.
    Nørsett, S.P., Thomsen, P.G.: Embedded SDIRK-methods of basic order three. BIT Numer. Math. 24(4), 634–646 (1984)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Palmer, K.: The structurally stable systems on the half-line are those with exponential dichotomy. J. Differ. Equ. 33(1), 16–25 (1979)CrossRefMATHGoogle Scholar
  36. 36.
    Perron, O.: Die stabilitätsfrage bei Differentialgleichungen. Math. Z. 32(1), 703–728 (1930)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Pötzsche, C.: Fine structure of the dichotomy spectrum. Integr. Equ. Oper. Theory 73(1), 107–151 (2012)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Pötzsche, C.: Dichotomy spectra of triangular equations. Discrete Contin. Dyn. Syst. Ser. A 36(1), 423–450 (2013)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Sacker, R., Sell, G.: A spectral theory for linear differential systems. J. Differ. Equ. 27(3), 320–358 (1978)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Steyer, A., Van Vleck, E.S.: Underlying one-step methods and nonautonomous stability of general linear methods. Discrete Contin. Dyn. Syst. Ser. B (2017).  https://doi.org/10.3934/dcdsb.2018108
  41. 41.
    Van der Pol, B.: A theory of the amplitude of free and forced triode vibration. Radio Rev. 1, 701–710 (1920)Google Scholar
  42. 42.
    Van Vleck, E.S.: On the error in the product QR decomposition. SIAM J. Matrix Anal. Appl. 31(4), 1775–1791 (2010)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Zenisek, A.: Nonlinear Elliptic and Evolution Problems and their Finite Element Approximations. Academic Press, London (1990)MATHGoogle Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Sandia National LaboratoriesAlbuquerqueUSA
  2. 2.Department of MathematicsUniversity of KansasLawrenceUSA

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