Year: 2017
Author: Carmen Arévalo, Gustaf Söderlind
Journal of Computational Mathematics, Vol. 35 (2017), Iss. 5 : pp. 672–692
Abstract
A new polynomial formulation of variable step size linear multistep methods is presented, where each k-step method is characterized by a fixed set of $k-1$ or $k$ parameters. This construction includes all methods of maximal order ($p=k$ for stiff, and $p=k+1$ for nonstiff problems). Supporting time step adaptivity by construction, the new formulation is not based on extending classical fixed step size methods; instead classical methods are obtained as fixed step size restrictions within a unified framework. The methods are implemented in Matlab, with local error estimation and a wide range of step size controllers. This provides a platform for investigating and comparing different multistep method in realistic operational conditions. Computational experiments show that the new multistep method construction and implementation compares favorably to existing software, although variable order has not yet been included.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.1611-m2015-0404
Journal of Computational Mathematics, Vol. 35 (2017), Iss. 5 : pp. 672–692
Published online: 2017-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 21
Keywords: Linear multistep methods Variable step size Adaptive step size Step size control Explicit methods Implicit methods Nonstiff methods Stiff methods Initial value problems Ordinary differential equations Differential-algebraic equations Implementation.
Author Details
-
A time adaptive multirate Dirichlet–Neumann waveform relaxation method for heterogeneous coupled heat equations
Meisrimel, Peter | Monge, Azahar | Birken, PhilippZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 103 (2023), Iss. 11
https://doi.org/10.1002/zamm.202100328 [Citations: 0] -
On the zero-stability of multistep methods on smooth nonuniform grids
Söderlind, Gustaf | Fekete, Imre | Faragó, IstvánBIT Numerical Mathematics, Vol. 58 (2018), Iss. 4 P.1125
https://doi.org/10.1007/s10543-018-0716-y [Citations: 8] -
Local error estimation and step size control in adaptive linear multistep methods
Arévalo, Carmen | Söderlind, Gustaf | Hadjimichael, Yiannis | Fekete, ImreNumerical Algorithms, Vol. 86 (2021), Iss. 2 P.537
https://doi.org/10.1007/s11075-020-00900-1 [Citations: 7] -
A Polynomial Formulation of Adaptive Strong Stability Preserving Multistep Methods
Mohammadi, Fatemeh | Arévalo, Carmen | Führer, ClausSIAM Journal on Numerical Analysis, Vol. 57 (2019), Iss. 1 P.27
https://doi.org/10.1137/17M1158811 [Citations: 1] -
General relaxation methods for initial-value problems with application to multistep schemes
Ranocha, Hendrik | Lóczi, Lajos | Ketcheson, David I.Numerische Mathematik, Vol. 146 (2020), Iss. 4 P.875
https://doi.org/10.1007/s00211-020-01158-4 [Citations: 28] -
A Software Platform for Adaptive High Order Multistep Methods
Arévalo, Carmen | Jonsson-Glans, Erik | Olander, Josefine | Soto, Monica Selva | Söderlind, GustafACM Transactions on Mathematical Software, Vol. 46 (2020), Iss. 1 P.1
https://doi.org/10.1145/3372159 [Citations: 1]