Construction and study of high-order accurate schemes for solving the one-dimensional heat equation
The method of undetermined coefficients on multipoint stencils with two time levels was used to construct compact difference schemes of O (τ 3 , h 6 ) accuracy intended for solving boundary value problems for the one-dimensional heat equation. The schemes were examined for von Neumann stability, and...
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| Veröffentlicht in: | Computational mathematics and mathematical physics 2014-07, Vol.54 (7), p.1110-1121, Article 1110 |
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| container_title | Computational mathematics and mathematical physics |
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| creator | Komarov, S. Yu Shapeev, V. P. |
| description | The method of undetermined coefficients on multipoint stencils with two time levels was used to construct compact difference schemes of
O
(τ
3
,
h
6
) accuracy intended for solving boundary value problems for the one-dimensional heat equation. The schemes were examined for von Neumann stability, and numerical experiments were conducted on a sequence of grids with mesh sizes tending to zero. One of the schemes was proved to be absolutely stable. It was shown that, for smooth solutions, the high order of convergence of the numerical solution agrees with the order of accuracy; moreover, solutions accurate up to ∼10
−12
are obtained on grids with spatial mesh sizes of ∼10
−2
. The formulas for the schemes are rather simple and easy to implement on a computer. |
| doi_str_mv | 10.1134/S0965542514070082 |
| format | Article |
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O
(τ
3
,
h
6
) accuracy intended for solving boundary value problems for the one-dimensional heat equation. The schemes were examined for von Neumann stability, and numerical experiments were conducted on a sequence of grids with mesh sizes tending to zero. One of the schemes was proved to be absolutely stable. It was shown that, for smooth solutions, the high order of convergence of the numerical solution agrees with the order of accuracy; moreover, solutions accurate up to ∼10
−12
are obtained on grids with spatial mesh sizes of ∼10
−2
. The formulas for the schemes are rather simple and easy to implement on a computer.</description><identifier>ISSN: 0965-5425</identifier><identifier>EISSN: 1555-6662</identifier><identifier>DOI: 10.1134/S0965542514070082</identifier><language>eng</language><publisher>Moscow: Pleiades Publishing</publisher><subject>Accuracy ; Algorithms ; Boundary value problems ; Computational mathematics ; Computational Mathematics and Numerical Analysis ; Computer simulation ; Construction ; Convergence ; Data exchange ; Heat equations ; Linear algebra ; Mathematical analysis ; Mathematical models ; Mathematics ; Mathematics and Statistics ; Methods ; Numerical analysis ; Simulation ; Stencils ; Studies</subject><ispartof>Computational mathematics and mathematical physics, 2014-07, Vol.54 (7), p.1110-1121, Article 1110</ispartof><rights>Pleiades Publishing, Ltd. 2014</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c349t-e59a756a2b9b5a2c51e836e5055e2ead048d88a5050b172a53c5d2ee6d80410e3</citedby><cites>FETCH-LOGICAL-c349t-e59a756a2b9b5a2c51e836e5055e2ead048d88a5050b172a53c5d2ee6d80410e3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1134/S0965542514070082$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1134/S0965542514070082$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27903,27904,41467,42536,51297</link.rule.ids></links><search><creatorcontrib>Komarov, S. Yu</creatorcontrib><creatorcontrib>Shapeev, V. P.</creatorcontrib><title>Construction and study of high-order accurate schemes for solving the one-dimensional heat equation</title><title>Computational mathematics and mathematical physics</title><addtitle>Comput. Math. and Math. Phys</addtitle><description>The method of undetermined coefficients on multipoint stencils with two time levels was used to construct compact difference schemes of
O
(τ
3
,
h
6
) accuracy intended for solving boundary value problems for the one-dimensional heat equation. The schemes were examined for von Neumann stability, and numerical experiments were conducted on a sequence of grids with mesh sizes tending to zero. One of the schemes was proved to be absolutely stable. It was shown that, for smooth solutions, the high order of convergence of the numerical solution agrees with the order of accuracy; moreover, solutions accurate up to ∼10
−12
are obtained on grids with spatial mesh sizes of ∼10
−2
. The formulas for the schemes are rather simple and easy to implement on a computer.</description><subject>Accuracy</subject><subject>Algorithms</subject><subject>Boundary value problems</subject><subject>Computational mathematics</subject><subject>Computational Mathematics and Numerical Analysis</subject><subject>Computer simulation</subject><subject>Construction</subject><subject>Convergence</subject><subject>Data exchange</subject><subject>Heat equations</subject><subject>Linear algebra</subject><subject>Mathematical analysis</subject><subject>Mathematical models</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Methods</subject><subject>Numerical analysis</subject><subject>Simulation</subject><subject>Stencils</subject><subject>Studies</subject><issn>0965-5425</issn><issn>1555-6662</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2014</creationdate><recordtype>article</recordtype><sourceid>BENPR</sourceid><recordid>eNp9kMtOwzAQRS0EEqXwAewssWET8CNjJ0tU8ZIqsQDWketMmlRp3NoOUv-ehLJARbCyRnPO9egScsnZDecyvX1luQJIBfCUacYycUQmHAASpZQ4JpNxnYz7U3IWwooxrvJMToiduS5E39vYuI6arqQh9uWOuorWzbJOnC_RU2Nt701EGmyNawy0cp4G13403ZLGGqnrMCmbNXZhiDEtrdFEitvejLHn5KQybcCL73dK3h_u32ZPyfzl8Xl2N0-sTPOYIORGgzJikS_ACAscM6kQGAAKNCVLszLLzDCzBdfCgLRQCkRVZizlDOWUXO9zN95tewyxWDfBYtuaDl0fCg5KMy0B9IBeHaAr1_vh8pFKNTCZpXKg-J6y3oXgsSo2vlkbvys4K8bai1-1D44-cGwTv2qI3jTtv6bYm2H4pVui_3HTn9IncriVUQ</recordid><startdate>20140701</startdate><enddate>20140701</enddate><creator>Komarov, S. 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Yu ; Shapeev, V. P.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c349t-e59a756a2b9b5a2c51e836e5055e2ead048d88a5050b172a53c5d2ee6d80410e3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2014</creationdate><topic>Accuracy</topic><topic>Algorithms</topic><topic>Boundary value problems</topic><topic>Computational mathematics</topic><topic>Computational Mathematics and Numerical Analysis</topic><topic>Computer simulation</topic><topic>Construction</topic><topic>Convergence</topic><topic>Data exchange</topic><topic>Heat equations</topic><topic>Linear algebra</topic><topic>Mathematical analysis</topic><topic>Mathematical models</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Methods</topic><topic>Numerical analysis</topic><topic>Simulation</topic><topic>Stencils</topic><topic>Studies</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Komarov, S. 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O
(τ
3
,
h
6
) accuracy intended for solving boundary value problems for the one-dimensional heat equation. The schemes were examined for von Neumann stability, and numerical experiments were conducted on a sequence of grids with mesh sizes tending to zero. One of the schemes was proved to be absolutely stable. It was shown that, for smooth solutions, the high order of convergence of the numerical solution agrees with the order of accuracy; moreover, solutions accurate up to ∼10
−12
are obtained on grids with spatial mesh sizes of ∼10
−2
. The formulas for the schemes are rather simple and easy to implement on a computer.</abstract><cop>Moscow</cop><pub>Pleiades Publishing</pub><doi>10.1134/S0965542514070082</doi><tpages>12</tpages></addata></record> |
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| source | Springer Nature - Complete Springer Journals |
| subjects | Accuracy Algorithms Boundary value problems Computational mathematics Computational Mathematics and Numerical Analysis Computer simulation Construction Convergence Data exchange Heat equations Linear algebra Mathematical analysis Mathematical models Mathematics Mathematics and Statistics Methods Numerical analysis Simulation Stencils Studies |
| title | Construction and study of high-order accurate schemes for solving the one-dimensional heat equation |
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