Analysis of Dual Consistency for Discontinuous Galerkin Discretizations of Source Terms
The effects of dual consistency on discontinuous Galerkin (DG) 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 sourc...
| Main Authors: | , |
|---|---|
| Format: | Technical Report |
| Language: | en_US |
| Published: |
Aerospace Computational Design Laboratory, Dept. of Aeronautics & Astronautics, Massachusetts Institute of Technology
2010
|
| Online Access: | http://hdl.handle.net/1721.1/57607 |
| _version_ | 1826206348475367424 |
|---|---|
| author | Oliver, Todd A. Darmofal, David L. |
| author_facet | Oliver, Todd A. Darmofal, David L. |
| author_sort | Oliver, Todd A. |
| collection | MIT |
| description | The effects of dual consistency on discontinuous Galerkin (DG) 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. A priori error estimates are derived to reveal the effect of dual
inconsistent discretization on computed functional outputs. Combined with bounds on the dual
consistency error, these estimates show that for a dual consistent discretization or the asymptotically
dual consistent discretization resulting from the mixed formulation, O(h2p) convergence
can be shown for linear problems and linear outputs. For similar but dual inconsistent schemes,
only O(hp) can be shown. 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 | 2024-09-23T13:28:05Z |
| format | Technical Report |
| id | mit-1721.1/57607 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T13:28:05Z |
| publishDate | 2010 |
| publisher | Aerospace Computational Design Laboratory, Dept. of Aeronautics & Astronautics, Massachusetts Institute of Technology |
| record_format | dspace |
| spelling | mit-1721.1/576072019-04-10T10:25:23Z Analysis of Dual Consistency for Discontinuous Galerkin Discretizations of Source Terms Oliver, Todd A. Darmofal, David L. The effects of dual consistency on discontinuous Galerkin (DG) 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. A priori error estimates are derived to reveal the effect of dual inconsistent discretization on computed functional outputs. Combined with bounds on the dual consistency error, these estimates show that for a dual consistent discretization or the asymptotically dual consistent discretization resulting from the mixed formulation, O(h2p) convergence can be shown for linear problems and linear outputs. For similar but dual inconsistent schemes, only O(hp) can be shown. 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. This work was supported by the U. S. Air Force Research Laboratory (USAF-3306-03-SC-0001) and The Boeing Company. 2010-08-27T20:06:02Z 2010-08-27T20:06:02Z 2006-09 Technical Report http://hdl.handle.net/1721.1/57607 en_US ACDL Technical Reports;ACDL TR-06-1 application/pdf Aerospace Computational Design Laboratory, Dept. of Aeronautics & Astronautics, Massachusetts Institute of Technology |
| spellingShingle | Oliver, Todd A. 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/1721.1/57607 |
| work_keys_str_mv | AT olivertodda analysisofdualconsistencyfordiscontinuousgalerkindiscretizationsofsourceterms AT darmofaldavidl analysisofdualconsistencyfordiscontinuousgalerkindiscretizationsofsourceterms |