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

Andrew J. Steyer; Erik S. Van Vleck

doi:10.3934/dcdsb.2018108

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2018, Volume 23, Issue 7: 2859-2877. Doi: 10.3934/dcdsb.2018108

This issue Previous Article Polynomial interpolation and a priori bootstrap for computer-assisted proofs in nonlinear ODEs Next Article Conditioning and relative error propagation in linear autonomous ordinary differential equations

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

Andrew J. Steyer^1, , and
Erik S. Van Vleck^2,

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
* Corresponding author: Andrew J. Steyer

Received: March 31, 2017

Revised: August 31, 2017

Early access: March 2018

Published: July 2018

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 65L05, 65L06; Secondary: 65L07, 65P40.

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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 $x(0)=(1,0)^T$.

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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 $x(0)=(1,0)^T$.

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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	$\mu_{\rm{appr}}(N_f/2,N_f/2)$
$7.5E-1$	$1.37E10$	$1.51E11$	$7.68E-1$
$7.5E-2$	$3.75E-3$	$9.42E-3$	$9.03E-3$
$7.5E-3$	$3.60E-7$	$6.38E-4$	$-9.70E-2$
$7.5E-4$	$1.95E-9$	$6.24E-5$	$-9.04E-2$

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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	$\mu_{\rm{appr}}(N_f/2,N_f/2)$	$\tau_{\rm{max}}$
$1.15$	$5.50E-5$	$4.38E-3$	$-2.33E-2$	$1.068$
$1.45$	$1.18E-4$	$5.02E-3$	$-1.69E-3$	$1.086$
$1.75$	$2.88E-4$	$5.70E-3$	$1.78E-2$	$1.11$
$2.05$	$7.96E-4$	$6.4E-3$	$3.64E-2$	$1.23$

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