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

  • * Corresponding author: Andrew J. Steyer

    * Corresponding author: Andrew J. Steyer 

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

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  • 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.

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

    Citation:

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

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