Analysis of Dual Consistency for Discontinuous Galerkin Discretizations of Source Terms
The effects of dual consistency on discontinuous Galerkin discretizations of solution and solution gradient dependent source terms are examined. Two common discretizations are analyzed: the standard weighting technique for source terms and the mixed formulation. It is shown that if the source term d...
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| Format: | Article |
| Language: | en_US |
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Society for Industrial and Applied Mathematics
2010
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| Online Access: | http://hdl.handle.net/1721.1/52399 |
| _version_ | 1826205043456475136 |
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| author | Darmofal, David L. |
| author2 | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics |
| author_facet | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Darmofal, David L. |
| author_sort | Darmofal, David L. |
| collection | MIT |
| description | The effects of dual consistency on discontinuous Galerkin discretizations of solution and solution gradient dependent source terms are examined. Two common discretizations are analyzed: the standard weighting technique for source terms and the mixed formulation. It is shown that if the source term depends on the first derivative of the solution, the standard weighting technique leads to a dual inconsistent scheme. A straightforward procedure for correcting this dual inconsistency and arriving at a dual consistent discretization is demonstrated. The mixed formulation, where the solution gradient in the source term is replaced by an additional variable that is solved for simultaneously with the state, leads to an asymptotically dual consistent discretization. Numerical results for a one-dimensional test problem confirm that the dual consistent and asymptotically dual consistent schemes achieve higher asymptotic convergence rates with grid refinement than a similar dual inconsistent scheme for both the primal and adjoint solutions as well as a simple functional output. |
| first_indexed | 2024-09-23T13:06:11Z |
| format | Article |
| id | mit-1721.1/52399 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T13:06:11Z |
| publishDate | 2010 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | mit-1721.1/523992022-09-28T12:00:41Z Analysis of Dual Consistency for Discontinuous Galerkin Discretizations of Source Terms Darmofal, David L. Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Darmofal, David L. Darmofal, David L. The effects of dual consistency on discontinuous Galerkin discretizations of solution and solution gradient dependent source terms are examined. Two common discretizations are analyzed: the standard weighting technique for source terms and the mixed formulation. It is shown that if the source term depends on the first derivative of the solution, the standard weighting technique leads to a dual inconsistent scheme. A straightforward procedure for correcting this dual inconsistency and arriving at a dual consistent discretization is demonstrated. The mixed formulation, where the solution gradient in the source term is replaced by an additional variable that is solved for simultaneously with the state, leads to an asymptotically dual consistent discretization. Numerical results for a one-dimensional test problem confirm that the dual consistent and asymptotically dual consistent schemes achieve higher asymptotic convergence rates with grid refinement than a similar dual inconsistent scheme for both the primal and adjoint solutions as well as a simple functional output. Boeing Company U. S. Air Force Research Laboratory (USAF-3306-03-SC-0001) 2010-03-08T21:09:36Z 2010-03-08T21:09:36Z 2009-11 2008-04 Article http://purl.org/eprint/type/JournalArticle 0036-1429 http://hdl.handle.net/1721.1/52399 Oliver, Todd A., and David L. Darmofal. “Analysis of Dual Consistency for Discontinuous Galerkin Discretizations of Source Terms.” SIAM Journal on Numerical Analysis 47.5 (2009): 3507-3525. ©2009 Society for Industrial and Applied Mathematics en_US http://dx.doi.org/10.1137/080721467 SIAM Journal on Numerical Analysis Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf Society for Industrial and Applied Mathematics SIAM |
| spellingShingle | Darmofal, David L. Analysis of Dual Consistency for Discontinuous Galerkin Discretizations of Source Terms |
| title | Analysis of Dual Consistency for Discontinuous Galerkin Discretizations of Source Terms |
| title_full | Analysis of Dual Consistency for Discontinuous Galerkin Discretizations of Source Terms |
| title_fullStr | Analysis of Dual Consistency for Discontinuous Galerkin Discretizations of Source Terms |
| title_full_unstemmed | Analysis of Dual Consistency for Discontinuous Galerkin Discretizations of Source Terms |
| title_short | Analysis of Dual Consistency for Discontinuous Galerkin Discretizations of Source Terms |
| title_sort | analysis of dual consistency for discontinuous galerkin discretizations of source terms |
| url | http://hdl.handle.net/1721.1/52399 |
| work_keys_str_mv | AT darmofaldavidl analysisofdualconsistencyfordiscontinuousgalerkindiscretizationsofsourceterms |