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

## 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. Here, 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.

- Authors:

- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
- Univ. of Kansas, Lawrence, KS (United States)

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA); National Science Foundation (NSF)

- OSTI Identifier:
- 1467393

- Report Number(s):
- SAND-2018-3998J

Journal ID: ISSN 1553-524X; 662444

- Grant/Contract Number:
- AC04-94AL85000; DMS-1419047

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Discrete & Continuous Dynamical Systems - B

- Additional Journal Information:
- Journal Volume: 23; Journal Issue: 7; Journal ID: ISSN 1553-524X

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING

### Citation Formats

```
J. Steyer, Andrew, and S. Van Vleck, Erik. Underlying one-step methods and nonautonomous stability of general linear methods. United States: N. p., 2018.
Web. doi:10.3934/dcdsb.2018108.
```

```
J. Steyer, Andrew, & S. Van Vleck, Erik. Underlying one-step methods and nonautonomous stability of general linear methods. United States. doi:10.3934/dcdsb.2018108.
```

```
J. Steyer, Andrew, and S. Van Vleck, Erik. Sat .
"Underlying one-step methods and nonautonomous stability of general linear methods". United States. doi:10.3934/dcdsb.2018108. https://www.osti.gov/servlets/purl/1467393.
```

```
@article{osti_1467393,
```

title = {Underlying one-step methods and nonautonomous stability of general linear methods},

author = {J. Steyer, Andrew and S. Van Vleck, Erik},

abstractNote = {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. Here, 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.},

doi = {10.3934/dcdsb.2018108},

journal = {Discrete & Continuous Dynamical Systems - B},

number = 7,

volume = 23,

place = {United States},

year = {2018},

month = {9}

}

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Cited by: 1 work

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