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.

[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.

[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.

[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.

[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.

[6]

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

[7]

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

[8]

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

[9]

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

[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.

[11]

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

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[21]

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

[22]

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

[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.

[24]

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

[25]

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

[26]

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

[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.

[28]

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

[29]

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

[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.

[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.

[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.

[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.

[34]

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

[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.

[36]

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

[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.

[38]

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

[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.

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.

[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.

[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.

[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.

[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.

[6]

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

[7]

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

[8]

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

[9]

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

[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.

[11]

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

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[21]

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

[22]

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

[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.

[24]

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

[25]

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

[26]

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

[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.

[28]

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

[29]

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

[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.

[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.

[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.

[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.

[34]

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

[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.

[36]

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

[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.

[38]

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

[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.

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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