Invariant Discretization Schemes Using Evolution-Projection Techniques
Finite difference discretization schemes preserving a subgroup of the maximal Lie invariance group of the one-dimensional linear heat equation are determined. These invariant schemes are constructed using the invariantization procedure for non-invariant schemes of the heat equation in computational...
Saved in:
| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Date: | 2013 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2013
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/149346 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Invariant Discretization Schemes Using Evolution-Projection Techniques / A. Bihlo, J. Nave // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 35 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862627833026707456 |
|---|---|
| author | Bihlo, A. Nave, J. |
| author_facet | Bihlo, A. Nave, J. |
| citation_txt | Invariant Discretization Schemes Using Evolution-Projection Techniques / A. Bihlo, J. Nave // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 35 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Finite difference discretization schemes preserving a subgroup of the maximal Lie invariance group of the one-dimensional linear heat equation are determined. These invariant schemes are constructed using the invariantization procedure for non-invariant schemes of the heat equation in computational coordinates. We propose a new methodology for handling moving discretization grids which are generally indispensable for invariant numerical schemes. The idea is to use the invariant grid equation, which determines the locations of the grid point at the next time level only for a single integration step and then to project the obtained solution to the regular grid using invariant interpolation schemes. This guarantees that the scheme is invariant and allows one to work on the simpler stationary grids. The discretization errors of the invariant schemes are established and their convergence rates are estimated. Numerical tests are carried out to shed some light on the numerical properties of invariant discretization schemes using the proposed evolution-projection strategy.
|
| first_indexed | 2025-12-07T13:39:50Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-149346 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T13:39:50Z |
| publishDate | 2013 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bihlo, A. Nave, J. 2019-02-21T07:06:03Z 2019-02-21T07:06:03Z 2013 Invariant Discretization Schemes Using Evolution-Projection Techniques / A. Bihlo, J. Nave // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 35 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 65M06; 58J70; 35K05 DOI: http://dx.doi.org/10.3842/SIGMA.2013.052 https://nasplib.isofts.kiev.ua/handle/123456789/149346 Finite difference discretization schemes preserving a subgroup of the maximal Lie invariance group of the one-dimensional linear heat equation are determined. These invariant schemes are constructed using the invariantization procedure for non-invariant schemes of the heat equation in computational coordinates. We propose a new methodology for handling moving discretization grids which are generally indispensable for invariant numerical schemes. The idea is to use the invariant grid equation, which determines the locations of the grid point at the next time level only for a single integration step and then to project the obtained solution to the regular grid using invariant interpolation schemes. This guarantees that the scheme is invariant and allows one to work on the simpler stationary grids. The discretization errors of the invariant schemes are established and their convergence rates are estimated. Numerical tests are carried out to shed some light on the numerical properties of invariant discretization schemes using the proposed evolution-projection strategy. This paper is a contribution to the Special Issue “Symmetries of Dif ferential Equations: Frames, Invariants
 and Applications”. The full collection is available at http://www.emis.de/journals/SIGMA/SDE2012.html.
 The authors thank Professor Roman Popovych for valuable discussions and careful reading of
 the manuscript. The valuable remarks of the anonymous referees are much appreciated. This
 research was supported by the Austrian Science Fund (FWF), project J3182–N13 (AB). JCN
 wishes to acknowledge partial support from the NSERC Discovery Program, and the National
 Science Foundation through grant DMS-0813648. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Invariant Discretization Schemes Using Evolution-Projection Techniques Article published earlier |
| spellingShingle | Invariant Discretization Schemes Using Evolution-Projection Techniques Bihlo, A. Nave, J. |
| title | Invariant Discretization Schemes Using Evolution-Projection Techniques |
| title_full | Invariant Discretization Schemes Using Evolution-Projection Techniques |
| title_fullStr | Invariant Discretization Schemes Using Evolution-Projection Techniques |
| title_full_unstemmed | Invariant Discretization Schemes Using Evolution-Projection Techniques |
| title_short | Invariant Discretization Schemes Using Evolution-Projection Techniques |
| title_sort | invariant discretization schemes using evolution-projection techniques |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149346 |
| work_keys_str_mv | AT bihloa invariantdiscretizationschemesusingevolutionprojectiontechniques AT navej invariantdiscretizationschemesusingevolutionprojectiontechniques |