September  2018, 23(7): 2859-2877. doi: 10.3934/dcdsb.2018108

Underlying one-step methods and nonautonomous stability of general linear methods

1. 

Sandia National Laboratories, P.O. Box 5800, MS 1320, Albuquerque, NM 87185-1320, USA

2. 

Department of Mathematics - University of Kansas, 1460 Jayhawk Blvd, Lawrence, KS 66045-7594, USA

* Corresponding author: Andrew J. Steyer

Dedicated to the memory of Timo Eirola.

Received  April 2017 Revised  September 2017 Published  March 2018

Fund Project: This research was supported in part by NSF grant DMS-1419047

We generalize the theory of underlying one-step methods to strictly stable general linear methods (GLMs) solving nonautonomous ordinary differential equations (ODEs) that satisfy a global Lipschitz condition. We combine this theory with the Lyapunov and Sacker-Sell spectral stability theory for one-step methods developed in [34,35,36] to analyze the stability of a strictly stable GLM solving a nonautonomous linear ODE. These results are applied to develop a stability diagnostic for the solution of nonautonomous linear ODEs by strictly stable GLMs.

Citation: Andrew J. Steyer, Erik S. Van Vleck. Underlying one-step methods and nonautonomous stability of general linear methods. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2859-2877. doi: 10.3934/dcdsb.2018108
References:
[1]

B. Aulbach, The fundamental existence theorem on invariant fiber bundles, J. Differ. Equ. Appl., 3 (1998), 501-537. doi: 10.1080/10236199708808118. Google Scholar

[2]

B. AulbachC. Pötzsche and S. Siegmund, A smoothness theorem for invariant fiber bundles, J. Dynam. Differential Equations, 14 (2002), 519-547. doi: 10.1023/A:1016383031231. Google Scholar

[3]

B. Aulbach, M. Rasmussen and S. Siegmund, Invariant manifolds as pullback attractors of nonautonomous difference equations, in Proceedings of the Eighth International Conference on Difference Equations and Applications (eds. B. Aulbach, O. Dosly, S. Elaydi, G. Ladas), Chapman & Hall/CRC, Boca Raton, FL (2005), 23–37. Google Scholar

[4]

B. AulbachM. Rasmussen and S. Siegmund, Invariant manifolds as pullback attractors of nonautonomous differential equations, Discrete Contin. Dyn. Syst., 15 (2006), 579-596. doi: 10.3934/dcds.2006.15.579. Google Scholar

[5]

B. Aulbach and T. Wanner, Invariant foliations and decoupling nonautonomous difference equations, J. Difference Eq. Appl., 9 (2003), 459-472. doi: 10.1080/1023619031000076524. Google Scholar

[6]

W.-J. Beyn, On invariant close curves for one-step methods, Numer. Math., 51 (1987), 103-122. doi: 10.1007/BF01399697. Google Scholar

[7]

J. Butcher, The equivalence of algebraic stability and AN-stability, BIT, 27 (1987), 510-533. doi: 10.1007/BF01937275. Google Scholar

[8]

J. Butcher, The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods, Wiley-Interscience New York, NY, 1987. Google Scholar

[9]

W. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, Berlin-New York, 1978. doi: 10.1007/BFb0067780. Google Scholar

[10]

G. Dahlquist, Convergence and stability in the numerical integration of ordinary differential equations, Math. Scan., 4 (1956), 33-53. doi: 10.7146/math.scand.a-10454. Google Scholar

[11]

G. Dahlquist, Stability and error bounds in the numerical integration of ordinary differential equations, Kungl. Tekn. Högsk. Handl. Stockholm. 130 (1959), 87 pp. Google Scholar

[12]

G. Dahlquist, A special stability problem for linear multistep methods, BIT, 3 (1963), 27-43. doi: 10.1007/BF01963532. Google Scholar

[13]

R. D'AmbrosioE. Hairer and C. Zbinden, G-symplecticity implies conjugate-symplecticity of the underlying one-step method, BIT, 53 (2013), 867-872. doi: 10.1007/s10543-013-0437-1. Google Scholar

[14]

L. Dieci and E. S. Van Vleck, Computation of a few Lyapunov exponents for continuous and discrete dynamical systems, Appl. Numer. Math., 17 (1995), 275-291. doi: 10.1016/0168-9274(95)00033-Q. Google Scholar

[15]

L. Dieci and E. S. Van Vleck, Lyapunov spectral intervals: Theory and computation, SIAM J. Numer. Anal., 40 (2002), 516-542. doi: 10.1137/S0036142901392304. Google Scholar

[16]

L. Dieci and E. S. Van Vleck, On the error in computing Lyapunov exponents by QR Methods, Numer. Math., 101 (2005), 619-642. doi: 10.1007/s00211-005-0644-z. Google Scholar

[17]

L. Dieci and E. S. Van Vleck, Perturbation theory for approximation of Lyapunov exponents by QR methods, J. Dynam. Differential Equations, 18 (2006), 825-840. doi: 10.1007/s10884-006-9024-3. Google Scholar

[18]

L. Dieci and E. S. Van Vleck, Lyapunov and Sacker-Sell spectral intervals, J. Dynam. Differential Equations, 19 (2007), 265-293. doi: 10.1007/s10884-006-9030-5. Google Scholar

[19]

J. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Modern Phy., 57 (1985), 617-656. doi: 10.1103/RevModPhys.57.617. Google Scholar

[20]

T. Eirola, Invariant curves of one-step methods, BIT, 28 (1988), 113-122. doi: 10.1007/BF01934699. Google Scholar

[21]

T. Eirola and O. Nevanlinna, What do multistep methods approximate?, Numer. Math., 53 (1988), 559-569. doi: 10.1007/BF01397552. Google Scholar

[22]

E. Hairer, Conjugate-symplecticity of linear multistep methods, J. Comput. Math., 26 (2008), 657-659. Google Scholar

[23]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, 2006. doi: 10.1007/978-3-662-05018-7. Google Scholar

[24]

Z. Jackiewicz, General Linear Methods for Ordinary Differential Equations, John Wiley & Sons Inc., Hoboken, N. J., 2009. Google Scholar

[25]

U. Kirchgraber, Multistep methods are essentially one-step methods, Numer. Math., 48 (1986), 85-90. doi: 10.1007/BF01389443. Google Scholar

[26]

H.-O. Kreiss, Difference methods for stiff ordinary differential equations, SIAM J. Numer. Anal., 15 (1978), 21-58. doi: 10.1137/0715003. Google Scholar

[27]

G. Leonov and N. Kuznetsov, Time-varying linearization and the Perron effects, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 1079-1107. doi: 10.1142/S0218127407017732. Google Scholar

[28]

A. Lyapunov, The general problem of the stability of motion, Internat. J. Control, 55 (1992), 521-790. doi: 10.1080/00207179208934253. Google Scholar

[29]

O. Perron, Die stabilitätsfrage bei differentialgleichungen, Math. Z., 32 (1930), 703-728. doi: 10.1007/BF01194662. Google Scholar

[30]

C. Pötzsche, Fine structure of the dichotomy spectrum, Integral Equations and Operator Theory, 73 (2012), 107-151. doi: 10.1007/s00020-012-1959-7. Google Scholar

[31]

C. Pötzsche and M. Rasmussen, Computation of integral manifolds for Carathéodory differential equations, IMA J. Numer. Anal., 30 (2010), 401-430. doi: 10.1093/imanum/drn059. Google Scholar

[32]

R. Sacker and G. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8. Google Scholar

[33]

R. JohnsonK. Palmer and G. Sell, Ergodic properties of linear dynamical systems, SIAM J. Math. Anal., 18 (1987), 1-33. doi: 10.1137/0518001. Google Scholar

[34]

A. Steyer, A Lyapunov Exponent Based Stability Theory for Ordinary Differential Equation Initial Value Problem Solvers, Ph. D thesis, University of Kansas, 2016. Google Scholar

[35]

A. Steyer and E. S. Van Vleck, A step-size selection strategy for explicit Runge-Kutta methods based on Lyapunov exponent theory, J. Comp. Appld. Math., 292 (2016), 703-719. doi: 10.1016/j.cam.2015.03.056. Google Scholar

[36]

A. Steyer and E. S. Van Vleck, A Lyapunov and Sacker-Sell Spectral Stability Theory for One-Step Methods, Submitted for publication, 2017.Google Scholar

[37]

K. Nipp and D. Stoffer, Attractive invariant manifolds for maps: Existence, smoothness and continuous dependence on the map, Research report, Applied Mathematics, ETH-Zurich, (1992), 92-111. Google Scholar

[38]

D. Stoffer, General linear methods: Connection to one-step methods and invariant curves, Numer. Math., 64 (1993), 395-408. doi: 10.1007/BF01388696. Google Scholar

[39]

E. S. Van Vleck, On the error in the product QR decomposition, SIAM J. Matrix Anal. Appl., 31 (2009/2010), 1775-1791. doi: 10.1137/090761562. Google Scholar

show all references

References:
[1]

B. Aulbach, The fundamental existence theorem on invariant fiber bundles, J. Differ. Equ. Appl., 3 (1998), 501-537. doi: 10.1080/10236199708808118. Google Scholar

[2]

B. AulbachC. Pötzsche and S. Siegmund, A smoothness theorem for invariant fiber bundles, J. Dynam. Differential Equations, 14 (2002), 519-547. doi: 10.1023/A:1016383031231. Google Scholar

[3]

B. Aulbach, M. Rasmussen and S. Siegmund, Invariant manifolds as pullback attractors of nonautonomous difference equations, in Proceedings of the Eighth International Conference on Difference Equations and Applications (eds. B. Aulbach, O. Dosly, S. Elaydi, G. Ladas), Chapman & Hall/CRC, Boca Raton, FL (2005), 23–37. Google Scholar

[4]

B. AulbachM. Rasmussen and S. Siegmund, Invariant manifolds as pullback attractors of nonautonomous differential equations, Discrete Contin. Dyn. Syst., 15 (2006), 579-596. doi: 10.3934/dcds.2006.15.579. Google Scholar

[5]

B. Aulbach and T. Wanner, Invariant foliations and decoupling nonautonomous difference equations, J. Difference Eq. Appl., 9 (2003), 459-472. doi: 10.1080/1023619031000076524. Google Scholar

[6]

W.-J. Beyn, On invariant close curves for one-step methods, Numer. Math., 51 (1987), 103-122. doi: 10.1007/BF01399697. Google Scholar

[7]

J. Butcher, The equivalence of algebraic stability and AN-stability, BIT, 27 (1987), 510-533. doi: 10.1007/BF01937275. Google Scholar

[8]

J. Butcher, The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods, Wiley-Interscience New York, NY, 1987. Google Scholar

[9]

W. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, Berlin-New York, 1978. doi: 10.1007/BFb0067780. Google Scholar

[10]

G. Dahlquist, Convergence and stability in the numerical integration of ordinary differential equations, Math. Scan., 4 (1956), 33-53. doi: 10.7146/math.scand.a-10454. Google Scholar

[11]

G. Dahlquist, Stability and error bounds in the numerical integration of ordinary differential equations, Kungl. Tekn. Högsk. Handl. Stockholm. 130 (1959), 87 pp. Google Scholar

[12]

G. Dahlquist, A special stability problem for linear multistep methods, BIT, 3 (1963), 27-43. doi: 10.1007/BF01963532. Google Scholar

[13]

R. D'AmbrosioE. Hairer and C. Zbinden, G-symplecticity implies conjugate-symplecticity of the underlying one-step method, BIT, 53 (2013), 867-872. doi: 10.1007/s10543-013-0437-1. Google Scholar

[14]

L. Dieci and E. S. Van Vleck, Computation of a few Lyapunov exponents for continuous and discrete dynamical systems, Appl. Numer. Math., 17 (1995), 275-291. doi: 10.1016/0168-9274(95)00033-Q. Google Scholar

[15]

L. Dieci and E. S. Van Vleck, Lyapunov spectral intervals: Theory and computation, SIAM J. Numer. Anal., 40 (2002), 516-542. doi: 10.1137/S0036142901392304. Google Scholar

[16]

L. Dieci and E. S. Van Vleck, On the error in computing Lyapunov exponents by QR Methods, Numer. Math., 101 (2005), 619-642. doi: 10.1007/s00211-005-0644-z. Google Scholar

[17]

L. Dieci and E. S. Van Vleck, Perturbation theory for approximation of Lyapunov exponents by QR methods, J. Dynam. Differential Equations, 18 (2006), 825-840. doi: 10.1007/s10884-006-9024-3. Google Scholar

[18]

L. Dieci and E. S. Van Vleck, Lyapunov and Sacker-Sell spectral intervals, J. Dynam. Differential Equations, 19 (2007), 265-293. doi: 10.1007/s10884-006-9030-5. Google Scholar

[19]

J. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Modern Phy., 57 (1985), 617-656. doi: 10.1103/RevModPhys.57.617. Google Scholar

[20]

T. Eirola, Invariant curves of one-step methods, BIT, 28 (1988), 113-122. doi: 10.1007/BF01934699. Google Scholar

[21]

T. Eirola and O. Nevanlinna, What do multistep methods approximate?, Numer. Math., 53 (1988), 559-569. doi: 10.1007/BF01397552. Google Scholar

[22]

E. Hairer, Conjugate-symplecticity of linear multistep methods, J. Comput. Math., 26 (2008), 657-659. Google Scholar

[23]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, 2006. doi: 10.1007/978-3-662-05018-7. Google Scholar

[24]

Z. Jackiewicz, General Linear Methods for Ordinary Differential Equations, John Wiley & Sons Inc., Hoboken, N. J., 2009. Google Scholar

[25]

U. Kirchgraber, Multistep methods are essentially one-step methods, Numer. Math., 48 (1986), 85-90. doi: 10.1007/BF01389443. Google Scholar

[26]

H.-O. Kreiss, Difference methods for stiff ordinary differential equations, SIAM J. Numer. Anal., 15 (1978), 21-58. doi: 10.1137/0715003. Google Scholar

[27]

G. Leonov and N. Kuznetsov, Time-varying linearization and the Perron effects, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 1079-1107. doi: 10.1142/S0218127407017732. Google Scholar

[28]

A. Lyapunov, The general problem of the stability of motion, Internat. J. Control, 55 (1992), 521-790. doi: 10.1080/00207179208934253. Google Scholar

[29]

O. Perron, Die stabilitätsfrage bei differentialgleichungen, Math. Z., 32 (1930), 703-728. doi: 10.1007/BF01194662. Google Scholar

[30]

C. Pötzsche, Fine structure of the dichotomy spectrum, Integral Equations and Operator Theory, 73 (2012), 107-151. doi: 10.1007/s00020-012-1959-7. Google Scholar

[31]

C. Pötzsche and M. Rasmussen, Computation of integral manifolds for Carathéodory differential equations, IMA J. Numer. Anal., 30 (2010), 401-430. doi: 10.1093/imanum/drn059. Google Scholar

[32]

R. Sacker and G. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8. Google Scholar

[33]

R. JohnsonK. Palmer and G. Sell, Ergodic properties of linear dynamical systems, SIAM J. Math. Anal., 18 (1987), 1-33. doi: 10.1137/0518001. Google Scholar

[34]

A. Steyer, A Lyapunov Exponent Based Stability Theory for Ordinary Differential Equation Initial Value Problem Solvers, Ph. D thesis, University of Kansas, 2016. Google Scholar

[35]

A. Steyer and E. S. Van Vleck, A step-size selection strategy for explicit Runge-Kutta methods based on Lyapunov exponent theory, J. Comp. Appld. Math., 292 (2016), 703-719. doi: 10.1016/j.cam.2015.03.056. Google Scholar

[36]

A. Steyer and E. S. Van Vleck, A Lyapunov and Sacker-Sell Spectral Stability Theory for One-Step Methods, Submitted for publication, 2017.Google Scholar

[37]

K. Nipp and D. Stoffer, Attractive invariant manifolds for maps: Existence, smoothness and continuous dependence on the map, Research report, Applied Mathematics, ETH-Zurich, (1992), 92-111. Google Scholar

[38]

D. Stoffer, General linear methods: Connection to one-step methods and invariant curves, Numer. Math., 64 (1993), 395-408. doi: 10.1007/BF01388696. Google Scholar

[39]

E. S. Van Vleck, On the error in the product QR decomposition, SIAM J. Matrix Anal. Appl., 31 (2009/2010), 1775-1791. doi: 10.1137/090761562. Google Scholar

Figure 1.  Left: Logarithmic plot of the 2-norm of the local truncation error of the numerical solution versus time for various values of $h$. Right: Logarithmic plot of the 2-norm of the numerical solution versus time for various values of $h$. The parameter values used were $a_1=a_2=1.2$, $b_1 = -0.14$, $b_2=-0.15$, $\beta=10.0$, $\omega = 1$ with a final time of $t_f = 40$ and the initial condition $x(0)=(1,0)^T$.
Figure 2.  Left: Logarithmic plot of the 2-norm of the local truncation error of the numerical solution versus time for various values of $h$. Right: Logarithmic plot of the 2-norm of the numerical solution versus time for various values of $h$. The parameter values used were using $b_1 = -0.5$, $b_2=-.055$, $\beta=1.0$, $\omega = 1$, and a final time of $t_f = 100$ for various values of $a=a_1=a_2$ using the step-sizes $h=0.05$ and the initial condition $x(0)=(1,0)^T$.
Table 1.  Results of an experiment for the solution of (3) using BDF2, $a_1=a_2=1.2$, $b_1 = -0.14$, $b_2=-0.15$, $\beta=10.0$, $\omega = 1$, and a final time of $t_f = 40$ for various step-sizes $h$ and the initial condition $x(0)=(1,0)^T$. LTEmean is the mean local truncation error, LTEmax is the maximum local truncation error, and ${\mu_{\rm{appr}}(N_f/2,N_f/2)}$ is the value of (26) where $N_f$ is the final step of the approximation.
$h$ LTEmean LTEmax $\mu_{\rm{appr}}(N_f/2,N_f/2)$
$7.5E-1$ $1.37E10$ $1.51E11$ $7.68E-1$
$7.5E-2$ $3.75E-3$ $9.42E-3$ $9.03E-3$
$7.5E-3$ $3.60E-7$ $6.38E-4$ $-9.70E-2$
$7.5E-4$ $1.95E-9$ $6.24E-5$ $-9.04E-2$
$h$ LTEmean LTEmax $\mu_{\rm{appr}}(N_f/2,N_f/2)$
$7.5E-1$ $1.37E10$ $1.51E11$ $7.68E-1$
$7.5E-2$ $3.75E-3$ $9.42E-3$ $9.03E-3$
$7.5E-3$ $3.60E-7$ $6.38E-4$ $-9.70E-2$
$7.5E-4$ $1.95E-9$ $6.24E-5$ $-9.04E-2$
Table 2.  Results of an experiment for the solution of (3) using BDF2, using $b_1 = -0.5$, $b_2=-.055$, $\beta=1.0$, $\omega = 1$, and a final time of $t_f = 100$ for various values of $a=a_1=a_2$ using the step-sizes $h=0.05$ and the initial condition $x(0)=(1,0)^T$. LTEmean is the mean local truncation error, LTEmax is the maximum local truncation error, ${\mu_{\rm{appr}}(N_f/2,N_f/2)}$ is the value of (26) where $N_f$ is the final step of the approximation, and ${\tau_{\rm{max}}}$ is the maximum value of $\tau_n$ which denotes the quotient of the local truncation error at time-steps $n+1$ and $n$.
$a_1=a_2=a$ LTEmean LTEmax $\mu_{\rm{appr}}(N_f/2,N_f/2)$ $\tau_{\rm{max}}$
$1.15$ $5.50E-5$ $4.38E-3$ $-2.33E-2$ $1.068$
$1.45$ $1.18E-4$ $5.02E-3$ $-1.69E-3$ $1.086$
$1.75$ $2.88E-4$ $5.70E-3$ $1.78E-2$ $1.11$
$2.05$ $7.96E-4$ $6.4E-3$ $3.64E-2$ $1.23$
$a_1=a_2=a$ LTEmean LTEmax $\mu_{\rm{appr}}(N_f/2,N_f/2)$ $\tau_{\rm{max}}$
$1.15$ $5.50E-5$ $4.38E-3$ $-2.33E-2$ $1.068$
$1.45$ $1.18E-4$ $5.02E-3$ $-1.69E-3$ $1.086$
$1.75$ $2.88E-4$ $5.70E-3$ $1.78E-2$ $1.11$
$2.05$ $7.96E-4$ $6.4E-3$ $3.64E-2$ $1.23$
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