Calculus on surfaces with general closest point functions
The Closest Point Method for solving partial differential equations (PDEs) posed on surfaces was recently introduced by Ruuth and Merriman [J. Comput. Phys. 2008] and successfully applied to a variety of surface PDEs. In this paper we study the theoretical foundations of this method. The main idea i...
| প্রধান লেখক: | , |
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| বিন্যাস: | Journal article |
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2012
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| _version_ | 1826293440450658304 |
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| author | März, T Macdonald, C |
| author_facet | März, T Macdonald, C |
| author_sort | März, T |
| collection | OXFORD |
| description | The Closest Point Method for solving partial differential equations (PDEs) posed on surfaces was recently introduced by Ruuth and Merriman [J. Comput. Phys. 2008] and successfully applied to a variety of surface PDEs. In this paper we study the theoretical foundations of this method. The main idea is that surface differentials of a surface function can be replaced with Cartesian differentials of its closest point extension, i.e., its composition with a closest point function. We introduce a general class of these closest point functions (a subset of differentiable retractions), show that these are exactly the functions necessary to satisfy the above idea, and give a geometric characterization this class. Finally, we construct some closest point functions and demonstrate their effectiveness numerically on surface PDEs. |
| first_indexed | 2024-03-07T03:30:08Z |
| format | Journal article |
| id | oxford-uuid:ba6d3d20-7c2c-4ba1-ae48-92f9dcd0fe5c |
| institution | University of Oxford |
| last_indexed | 2024-03-07T03:30:08Z |
| publishDate | 2012 |
| record_format | dspace |
| spelling | oxford-uuid:ba6d3d20-7c2c-4ba1-ae48-92f9dcd0fe5c2022-03-27T05:09:44ZCalculus on surfaces with general closest point functionsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:ba6d3d20-7c2c-4ba1-ae48-92f9dcd0fe5cMathematical Institute - ePrints2012März, TMacdonald, CThe Closest Point Method for solving partial differential equations (PDEs) posed on surfaces was recently introduced by Ruuth and Merriman [J. Comput. Phys. 2008] and successfully applied to a variety of surface PDEs. In this paper we study the theoretical foundations of this method. The main idea is that surface differentials of a surface function can be replaced with Cartesian differentials of its closest point extension, i.e., its composition with a closest point function. We introduce a general class of these closest point functions (a subset of differentiable retractions), show that these are exactly the functions necessary to satisfy the above idea, and give a geometric characterization this class. Finally, we construct some closest point functions and demonstrate their effectiveness numerically on surface PDEs. |
| spellingShingle | März, T Macdonald, C Calculus on surfaces with general closest point functions |
| title | Calculus on surfaces with general closest point functions |
| title_full | Calculus on surfaces with general closest point functions |
| title_fullStr | Calculus on surfaces with general closest point functions |
| title_full_unstemmed | Calculus on surfaces with general closest point functions |
| title_short | Calculus on surfaces with general closest point functions |
| title_sort | calculus on surfaces with general closest point functions |
| work_keys_str_mv | AT marzt calculusonsurfaceswithgeneralclosestpointfunctions AT macdonaldc calculusonsurfaceswithgeneralclosestpointfunctions |