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Volume 35, Issue 5
Grid-Independent Construction of Multistep Methods

Carmen Arévalo & Gustaf Söderlind

DOI: 10.4208/jcm.1611-m2015-0404

J. Comp. Math., 35 (2017), pp. 672-692.

Published online: 2017-10

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  • 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.

  • 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.

  • AMS Subject Headings

65L06, 65L05, 65L80.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

Carmen.Arevalo@na.lu.se (Carmen Arévalo)

Gustaf.Soderlind@na.lu.se (Gustaf Söderlind)

  • BibTex
  • RIS
  • TXT
@Article{JCM-35-672, author = {Arévalo , Carmen and Söderlind , Gustaf}, title = {Grid-Independent Construction of Multistep Methods}, journal = {Journal of Computational Mathematics}, year = {2017}, volume = {35}, number = {5}, pages = {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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1611-m2015-0404}, url = {http://global-sci.org/intro/article_detail/jcm/10037.html} }
TY - JOUR T1 - Grid-Independent Construction of Multistep Methods AU - Arévalo , Carmen AU - Söderlind , Gustaf JO - Journal of Computational Mathematics VL - 5 SP - 672 EP - 692 PY - 2017 DA - 2017/10 SN - 35 DO - http://doi.org/10.4208/jcm.1611-m2015-0404 UR - https://global-sci.org/intro/article_detail/jcm/10037.html KW - 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. AB -

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.

Arévalo , Carmen and Söderlind , Gustaf. (2017). Grid-Independent Construction of Multistep Methods. Journal of Computational Mathematics. 35 (5). 672-692. doi:10.4208/jcm.1611-m2015-0404
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