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A software platform for adaptive high order multistep methods

Arévalo, Carmen LU ; Jonsson-Glans, Erik ; Olander, Josefine ; Soto, Monica Selva and Söderlind, Gustaf LU (2020) In ACM Transactions on Mathematical Software 46(1).
Abstract

We present a software package, Modes, offering h-adaptive and p-adaptive linear multistep methods for first order initial value problems in ordinary differential equations. The implementation is based on a new parametric, grid-independent representation of multistep methods [Arévalo and Söderlind 2017]. Parameters are supplied for over 60 methods. For nonstiff problems, all maximal order methods (p=k for explicit and p=k+1 for implicit methods) are supported. For stiff computation, implicit methods of order p=k are included. A collection of step-size controllers based on digital filters is provided, generating smooth step-size sequences offering improved computational stability. Controllers may be selected to match method and problem... (More)

We present a software package, Modes, offering h-adaptive and p-adaptive linear multistep methods for first order initial value problems in ordinary differential equations. The implementation is based on a new parametric, grid-independent representation of multistep methods [Arévalo and Söderlind 2017]. Parameters are supplied for over 60 methods. For nonstiff problems, all maximal order methods (p=k for explicit and p=k+1 for implicit methods) are supported. For stiff computation, implicit methods of order p=k are included. A collection of step-size controllers based on digital filters is provided, generating smooth step-size sequences offering improved computational stability. Controllers may be selected to match method and problem classes. A new system for automatic order control is also provided for designated families of multistep methods, offering simultaneous h- and p-adaptivity. Implemented as a Matlab toolbox, the software covers high order computations with linear multistep methods within a unified, generic framework. Computational experiments show that the new software is competitive and offers qualitative improvements. Modes is available for downloading and is primarily intended as a platform for developing a new generation of state-of-the-art multistep solvers, as well as for true ceteris paribus evaluation of algorithmic components. This also enables method comparisons within a single implementation environment.

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organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
multistep methods, ordinary differential equations, Solver, variable order, variable step size
in
ACM Transactions on Mathematical Software
volume
46
issue
1
article number
2
publisher
Association for Computing Machinery (ACM)
external identifiers
  • scopus:85084748673
ISSN
0098-3500
DOI
10.1145/3372159
language
English
LU publication?
yes
id
c2d60224-0d45-4d34-9011-936e0ed4b1f6
date added to LUP
2020-06-10 11:04:06
date last changed
2020-10-07 06:59:54
@article{c2d60224-0d45-4d34-9011-936e0ed4b1f6,
  abstract     = {<p>We present a software package, Modes, offering h-adaptive and p-adaptive linear multistep methods for first order initial value problems in ordinary differential equations. The implementation is based on a new parametric, grid-independent representation of multistep methods [Arévalo and Söderlind 2017]. Parameters are supplied for over 60 methods. For nonstiff problems, all maximal order methods (p=k for explicit and p=k+1 for implicit methods) are supported. For stiff computation, implicit methods of order p=k are included. A collection of step-size controllers based on digital filters is provided, generating smooth step-size sequences offering improved computational stability. Controllers may be selected to match method and problem classes. A new system for automatic order control is also provided for designated families of multistep methods, offering simultaneous h- and p-adaptivity. Implemented as a Matlab toolbox, the software covers high order computations with linear multistep methods within a unified, generic framework. Computational experiments show that the new software is competitive and offers qualitative improvements. Modes is available for downloading and is primarily intended as a platform for developing a new generation of state-of-the-art multistep solvers, as well as for true ceteris paribus evaluation of algorithmic components. This also enables method comparisons within a single implementation environment.</p>},
  author       = {Arévalo, Carmen and Jonsson-Glans, Erik and Olander, Josefine and Soto, Monica Selva and Söderlind, Gustaf},
  issn         = {0098-3500},
  language     = {eng},
  number       = {1},
  publisher    = {Association for Computing Machinery (ACM)},
  series       = {ACM Transactions on Mathematical Software},
  title        = {A software platform for adaptive high order multistep methods},
  url          = {http://dx.doi.org/10.1145/3372159},
  doi          = {10.1145/3372159},
  volume       = {46},
  year         = {2020},
}