Order conditions for General Linear Nyström methods
The purpose of this paper is to analyze the algebraic theory of order for the family of general linear Nyström (GLN) methods introduced in D’Ambrosio et al. (Numer. Algorithm 61 (2), 331–349, 2012 ) with the aim to provide a general framework for the representation and analysis of numerical methods...
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| Published in | Numerical algorithms Vol. 65; no. 3; pp. 579 - 595 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Boston
Springer US
01.03.2014
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1017-1398 1572-9265 |
| DOI | 10.1007/s11075-013-9819-3 |
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| Abstract | The purpose of this paper is to analyze the algebraic theory of order for the family of general linear Nyström (GLN) methods introduced in D’Ambrosio et al. (Numer. Algorithm
61
(2), 331–349,
2012
) with the aim to provide a general framework for the representation and analysis of numerical methods solving initial value problems based on second order ordinary differential equations (ODEs). Our investigation is carried out by suitably extending the theory of B-series for second order ODEs to the case of GLN methods, which leads to a general set of order conditions. This allows to recover the order conditions of numerical methods already known in the literature, but also to assess a general approach to study the order conditions of new methods, simply regarding them as GLN methods: the obtained results are indeed applied to both known and new methods for second order ODEs. |
|---|---|
| AbstractList | The purpose of this paper is to analyze the algebraic theory of order for the family of general linear Nystrom (GLN) methods introduced in D'Ambrosio et al. (Numer. Algorithm 61(2), 331-349, 2012) with the aim to provide a general framework for the representation and analysis of numerical methods solving initial value problems based on second order ordinary differential equations (ODEs). Our investigation is carried out by suitably extending the theory of B-series for second order ODEs to the case of GLN methods, which leads to a general set of order conditions. This allows to recover the order conditions of numerical methods already known in the literature, but also to assess a general approach to study the order conditions of new methods, simply regarding them as GLN methods: the obtained results are indeed applied to both known and new methods for second order ODEs. The purpose of this paper is to analyze the algebraic theory of order for the family of general linear Nyström (GLN) methods introduced in D’Ambrosio et al. (Numer. Algorithm 61 (2), 331–349, 2012 ) with the aim to provide a general framework for the representation and analysis of numerical methods solving initial value problems based on second order ordinary differential equations (ODEs). Our investigation is carried out by suitably extending the theory of B-series for second order ODEs to the case of GLN methods, which leads to a general set of order conditions. This allows to recover the order conditions of numerical methods already known in the literature, but also to assess a general approach to study the order conditions of new methods, simply regarding them as GLN methods: the obtained results are indeed applied to both known and new methods for second order ODEs. The purpose of this paper is to analyze the algebraic theory of order for the family of general linear Nyström (GLN) methods introduced in D’Ambrosio et al. (Numer. Algorithm 61(2), 331–349, 2012) with the aim to provide a general framework for the representation and analysis of numerical methods solving initial value problems based on second order ordinary differential equations (ODEs). Our investigation is carried out by suitably extending the theory of B-series for second order ODEs to the case of GLN methods, which leads to a general set of order conditions. This allows to recover the order conditions of numerical methods already known in the literature, but also to assess a general approach to study the order conditions of new methods, simply regarding them as GLN methods: the obtained results are indeed applied to both known and new methods for second order ODEs. |
| Author | De Martino, Giuseppe Paternoster, Beatrice D’Ambrosio, Raffaele |
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| CitedBy_id | crossref_primary_10_1007_s11075_016_0114_y crossref_primary_10_1007_s40314_020_01200_z crossref_primary_10_1016_j_matcom_2014_04_007 crossref_primary_10_3846_13926292_2015_1085921 crossref_primary_10_1088_1742_6596_1564_1_012012 crossref_primary_10_1080_00207160_2020_1737860 crossref_primary_10_1088_1742_6596_1564_1_012013 crossref_primary_10_1016_j_cam_2015_04_041 crossref_primary_10_1080_00207160_2018_1464154 |
| Cites_doi | 10.1016/S0168-9274(98)00056-7 10.1007/s10543-010-0279-z 10.1023/B:BITN.0000046809.66837.cc 10.1007/s11075-009-9280-5 10.1093/imanum/23.2.197 10.1016/S0168-9274(97)00032-9 10.1007/s002200050499 10.1016/j.cam.2012.08.012 10.1007/s10910-011-9903-7 10.1007/s11075-009-9329-5 10.1016/j.matcom.2011.01.005 10.1007/s11075-009-9285-0 10.1007/s10208-010-9065-1 10.1016/j.apnum.2012.06.008 10.1016/j.matcom.2010.10.011 10.1016/j.cam.2011.01.048 10.1007/BF01396335 10.1002/9780470753767 10.1016/S0898-1221(02)00136-0 10.1002/9780470522165 10.1016/j.aml.2009.01.017 10.1007/s11075-012-9637-z 10.1017/S0962492906220014 10.1007/3-540-47789-6_48 10.1023/A:1021921008328 10.1007/11758549_95 10.1007/3-540-44862-4_15 |
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| Keywords | 65L05 Order conditions Second order ordinary differential equations B-series General linear methods Nyström methods |
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| Snippet | The purpose of this paper is to analyze the algebraic theory of order for the family of general linear Nyström (GLN) methods introduced in D’Ambrosio et al.... The purpose of this paper is to analyze the algebraic theory of order for the family of general linear Nystrom (GLN) methods introduced in D'Ambrosio et al.... |
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| SubjectTerms | Algebra Algorithms Boundary value problems Computer Science Differential equations Economic conditions Initial value problems Numeric Computing Numerical Analysis Numerical methods Original Paper Representations Theory of Computation |
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| Title | Order conditions for General Linear Nyström methods |
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