Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers’ equation
In this paper we present rigorous a posteriori L 2 error bounds for reduced basis approximations of the unsteady viscous Burgers’ equation in one space dimension. The a posteriori error estimator, derived from standard analysis of the error-residual equation, comprises two key ingredients—both of wh...
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| Format: | Article |
| Language: | en_US |
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Springer-Verlag
2013
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| Online Access: | http://hdl.handle.net/1721.1/81221 https://orcid.org/0000-0002-0810-8812 https://orcid.org/0000-0002-2631-6463 |
| _version_ | 1826216417041580032 |
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| author | Nguyen, Ngoc Cuong Rozza, Gianluigi Patera, Anthony T. |
| author2 | Massachusetts Institute of Technology. Department of Mechanical Engineering |
| author_facet | Massachusetts Institute of Technology. Department of Mechanical Engineering Nguyen, Ngoc Cuong Rozza, Gianluigi Patera, Anthony T. |
| author_sort | Nguyen, Ngoc Cuong |
| collection | MIT |
| description | In this paper we present rigorous a posteriori L 2 error bounds for reduced basis approximations of the unsteady viscous Burgers’ equation in one space dimension. The a posteriori error estimator, derived from standard analysis of the error-residual equation, comprises two key ingredients—both of which admit efficient Offline-Online treatment: the first is a sum over timesteps of the square of the dual norm of the residual; the second is an accurate upper bound (computed by the Successive Constraint Method) for the exponential-in-time stability factor. These error bounds serve both Offline for construction of the reduced basis space by a new POD-Greedy procedure and Online for verification of fidelity. The a posteriori error bounds are practicable for final times (measured in convective units) T≈O(1) and Reynolds numbers ν[superscript −1]≫1; we present numerical results for a (stationary) steepening front for T=2 and 1≤ν[superscript −1]≤200. |
| first_indexed | 2024-09-23T16:47:21Z |
| format | Article |
| id | mit-1721.1/81221 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T16:47:21Z |
| publishDate | 2013 |
| publisher | Springer-Verlag |
| record_format | dspace |
| spelling | mit-1721.1/812212022-09-29T21:28:38Z Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers’ equation Nguyen, Ngoc Cuong Rozza, Gianluigi Patera, Anthony T. Massachusetts Institute of Technology. Department of Mechanical Engineering Nguyen, Ngoc Cuong Patera, Anthony T. Rozza, Gianluigi In this paper we present rigorous a posteriori L 2 error bounds for reduced basis approximations of the unsteady viscous Burgers’ equation in one space dimension. The a posteriori error estimator, derived from standard analysis of the error-residual equation, comprises two key ingredients—both of which admit efficient Offline-Online treatment: the first is a sum over timesteps of the square of the dual norm of the residual; the second is an accurate upper bound (computed by the Successive Constraint Method) for the exponential-in-time stability factor. These error bounds serve both Offline for construction of the reduced basis space by a new POD-Greedy procedure and Online for verification of fidelity. The a posteriori error bounds are practicable for final times (measured in convective units) T≈O(1) and Reynolds numbers ν[superscript −1]≫1; we present numerical results for a (stationary) steepening front for T=2 and 1≤ν[superscript −1]≤200. United States. Air Force Office of Scientific Research (AFOSR Grant FA9550-05-1-0114) United States. Air Force Office of Scientific Research (AFOSR Grant FA-9550-07-1-0425) Singapore-MIT Alliance for Research and Technology 2013-09-27T16:37:16Z 2013-09-27T16:37:16Z 2009-06 2009-04 Article http://purl.org/eprint/type/JournalArticle 0008-0624 1126-5434 http://hdl.handle.net/1721.1/81221 Nguyen, Ngoc-Cuong, Gianluigi Rozza, and Anthony T. Patera. Reduced Basis Approximation and a Posteriori Error Estimation for the Time-dependent Viscous Burgers’ Equation. Calcolo 46, no. 3 (September 30, 2009): 157-185. https://orcid.org/0000-0002-0810-8812 https://orcid.org/0000-0002-2631-6463 en_US http://dx.doi.org/10.1007/s10092-009-0005-x Calcolo Creative Commons Attribution-Noncommercial-Share Alike 3.0 http://creativecommons.org/licenses/by-nc-sa/3.0/ application/pdf Springer-Verlag MIT web domain |
| spellingShingle | Nguyen, Ngoc Cuong Rozza, Gianluigi Patera, Anthony T. Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers’ equation |
| title | Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers’ equation |
| title_full | Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers’ equation |
| title_fullStr | Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers’ equation |
| title_full_unstemmed | Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers’ equation |
| title_short | Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers’ equation |
| title_sort | reduced basis approximation and a posteriori error estimation for the time dependent viscous burgers equation |
| url | http://hdl.handle.net/1721.1/81221 https://orcid.org/0000-0002-0810-8812 https://orcid.org/0000-0002-2631-6463 |
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