Some observations on preconditioning for non-self-adjoint and time-dependent problems
Numerical Linear Algebra—specifically the computational solution of equations—forms a significant part of Computational Methods for Partial Differential Equations. Here we discuss the contrast between the solution of symmetric systems of equations that arise from self-adjoint problems and non-symmet...
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| Format: | Journal article |
| Language: | English |
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Elsevier
2021
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| _version_ | 1797108163223224320 |
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| author | Wathen, AJ |
| author_facet | Wathen, AJ |
| author_sort | Wathen, AJ |
| collection | OXFORD |
| description | Numerical Linear Algebra—specifically the computational solution of equations—forms a significant part of Computational Methods for Partial Differential Equations. Here we discuss the contrast between the solution of symmetric systems of equations that arise from self-adjoint problems and non-symmetric systems that arise from non-self-adjoint problems when iterative methods are employed; such methods are the only feasible methods for very large scale computation with PDEs. We then go on to consider non-symmetric all-at-once systems that arise in approximation of time-dependent problems, discuss causality and the parallel-in-time paradigm, suggesting an approach that involves preconditioning initial value problems with time-periodic problems. |
| first_indexed | 2024-03-07T07:25:37Z |
| format | Journal article |
| id | oxford-uuid:55693520-ac99-463e-b610-e8adf026720a |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:25:37Z |
| publishDate | 2021 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | oxford-uuid:55693520-ac99-463e-b610-e8adf026720a2022-11-14T09:21:18ZSome observations on preconditioning for non-self-adjoint and time-dependent problemsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:55693520-ac99-463e-b610-e8adf026720aEnglishSymplectic ElementsElsevier2021Wathen, AJNumerical Linear Algebra—specifically the computational solution of equations—forms a significant part of Computational Methods for Partial Differential Equations. Here we discuss the contrast between the solution of symmetric systems of equations that arise from self-adjoint problems and non-symmetric systems that arise from non-self-adjoint problems when iterative methods are employed; such methods are the only feasible methods for very large scale computation with PDEs. We then go on to consider non-symmetric all-at-once systems that arise in approximation of time-dependent problems, discuss causality and the parallel-in-time paradigm, suggesting an approach that involves preconditioning initial value problems with time-periodic problems. |
| spellingShingle | Wathen, AJ Some observations on preconditioning for non-self-adjoint and time-dependent problems |
| title | Some observations on preconditioning for non-self-adjoint and time-dependent problems |
| title_full | Some observations on preconditioning for non-self-adjoint and time-dependent problems |
| title_fullStr | Some observations on preconditioning for non-self-adjoint and time-dependent problems |
| title_full_unstemmed | Some observations on preconditioning for non-self-adjoint and time-dependent problems |
| title_short | Some observations on preconditioning for non-self-adjoint and time-dependent problems |
| title_sort | some observations on preconditioning for non self adjoint and time dependent problems |
| work_keys_str_mv | AT wathenaj someobservationsonpreconditioningfornonselfadjointandtimedependentproblems |