### Abstract

In order to be convergent, linear multistep methods must be zero stable. While constant step size theory was established in the 1950’s, zero stability on nonuniform grids is less well understood. Here we investigate zero stability on compact intervals and smooth nonuniform grids. In practical computations, step size control can be implemented using smooth (small) step size changes. The resulting grid (Formula presented.) can be modeled as the image of an equidistant grid under a smooth deformation map, i.e., (Formula presented.), where (Formula presented.) and the map (Formula presented.) is monotonically increasing with (Formula presented.) and (Formula presented.). The model is justified for any fixed order method operating in its asymptotic regime when applied to smooth problems, since the step size is then determined by the (smooth) principal error function which determines (Formula presented.), and a tolerance requirement which determines N. Given any strongly stable multistep method, there is an (Formula presented.) such that the method is zero stable for (Formula presented.), provided that (Formula presented.). Thus zero stability holds on all nonuniform grids such that adjacent step sizes satisfy (Formula presented.) as (Formula presented.). The results are exemplified for BDF-type methods.

Original language | English |
---|---|

Pages (from-to) | 1-19 |

Number of pages | 19 |

Journal | BIT Numerical Mathematics |

DOIs | |

Publication status | Accepted/In press - Jul 10 2018 |

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### Keywords

- BDF methods
- Convergence
- Initial value problems
- Linear multistep methods
- Nonuniform grids
- Variable step size
- Zero stability

### ASJC Scopus subject areas

- Software
- Computer Networks and Communications
- Computational Mathematics
- Applied Mathematics

### Cite this

*BIT Numerical Mathematics*, 1-19. https://doi.org/10.1007/s10543-018-0716-y

**On the zero-stability of multistep methods on smooth nonuniform grids.** / Söderlind, Gustaf; Fekete, Imre; Faragó, István.

Research output: Contribution to journal › Article

*BIT Numerical Mathematics*, pp. 1-19. https://doi.org/10.1007/s10543-018-0716-y

}

TY - JOUR

T1 - On the zero-stability of multistep methods on smooth nonuniform grids

AU - Söderlind, Gustaf

AU - Fekete, Imre

AU - Faragó, István

PY - 2018/7/10

Y1 - 2018/7/10

N2 - In order to be convergent, linear multistep methods must be zero stable. While constant step size theory was established in the 1950’s, zero stability on nonuniform grids is less well understood. Here we investigate zero stability on compact intervals and smooth nonuniform grids. In practical computations, step size control can be implemented using smooth (small) step size changes. The resulting grid (Formula presented.) can be modeled as the image of an equidistant grid under a smooth deformation map, i.e., (Formula presented.), where (Formula presented.) and the map (Formula presented.) is monotonically increasing with (Formula presented.) and (Formula presented.). The model is justified for any fixed order method operating in its asymptotic regime when applied to smooth problems, since the step size is then determined by the (smooth) principal error function which determines (Formula presented.), and a tolerance requirement which determines N. Given any strongly stable multistep method, there is an (Formula presented.) such that the method is zero stable for (Formula presented.), provided that (Formula presented.). Thus zero stability holds on all nonuniform grids such that adjacent step sizes satisfy (Formula presented.) as (Formula presented.). The results are exemplified for BDF-type methods.

AB - In order to be convergent, linear multistep methods must be zero stable. While constant step size theory was established in the 1950’s, zero stability on nonuniform grids is less well understood. Here we investigate zero stability on compact intervals and smooth nonuniform grids. In practical computations, step size control can be implemented using smooth (small) step size changes. The resulting grid (Formula presented.) can be modeled as the image of an equidistant grid under a smooth deformation map, i.e., (Formula presented.), where (Formula presented.) and the map (Formula presented.) is monotonically increasing with (Formula presented.) and (Formula presented.). The model is justified for any fixed order method operating in its asymptotic regime when applied to smooth problems, since the step size is then determined by the (smooth) principal error function which determines (Formula presented.), and a tolerance requirement which determines N. Given any strongly stable multistep method, there is an (Formula presented.) such that the method is zero stable for (Formula presented.), provided that (Formula presented.). Thus zero stability holds on all nonuniform grids such that adjacent step sizes satisfy (Formula presented.) as (Formula presented.). The results are exemplified for BDF-type methods.

KW - BDF methods

KW - Convergence

KW - Initial value problems

KW - Linear multistep methods

KW - Nonuniform grids

KW - Variable step size

KW - Zero stability

UR - http://www.scopus.com/inward/record.url?scp=85049665841&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85049665841&partnerID=8YFLogxK

U2 - 10.1007/s10543-018-0716-y

DO - 10.1007/s10543-018-0716-y

M3 - Article

SP - 1

EP - 19

JO - BIT Numerical Mathematics

JF - BIT Numerical Mathematics

SN - 0006-3835

ER -