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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...
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Інститут математики НАН України
2013
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
| Online Access: | http://dspace.nbuv.gov.ua/handle/123456789/149346 |
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irk-123456789-1493462019-02-23T01:23:27Z Invariant Discretization Schemes Using Evolution-Projection Techniques Bihlo, A. Nave, J. 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. 2013 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/149346 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| 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. |
| format |
Article |
| author |
Bihlo, A. Nave, J. |
| spellingShingle |
Bihlo, A. Nave, J. Invariant Discretization Schemes Using Evolution-Projection Techniques Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Bihlo, A. Nave, J. |
| author_sort |
Bihlo, A. |
| title |
Invariant Discretization Schemes Using Evolution-Projection Techniques |
| title_short |
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_sort |
invariant discretization schemes using evolution-projection techniques |
| publisher |
Інститут математики НАН України |
| publishDate |
2013 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/149346 |
| 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. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT bihloa invariantdiscretizationschemesusingevolutionprojectiontechniques AT navej invariantdiscretizationschemesusingevolutionprojectiontechniques |
| first_indexed |
2023-05-20T17:32:46Z |
| last_indexed |
2023-05-20T17:32:46Z |
| _version_ |
1796153535325274112 |