Convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws. Part 2: adjoint approximations and extensions
This paper continues the convergence analysis in [M. Giles and S. Ulbrich, SIAM J. Numer. Anal., 48 (2010), pp. 882–904] of discrete approximations to the linearized and adjoint equations arising from an unsteady one-dimensional hyperbolic equation with a convex flux function. We consider a simple m...
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| Format: | Journal article |
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SIAM
2010
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| _version_ | 1797061671268646912 |
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| author | Giles, M Ulbrich, S |
| author_facet | Giles, M Ulbrich, S |
| author_sort | Giles, M |
| collection | OXFORD |
| description | This paper continues the convergence analysis in [M. Giles and S. Ulbrich, SIAM J. Numer. Anal., 48 (2010), pp. 882–904] of discrete approximations to the linearized and adjoint equations arising from an unsteady one-dimensional hyperbolic equation with a convex flux function. We consider a simple modified Lax–Friedrichs discretization on a uniform grid, and a key point is that the numerical smoothing increases the number of points across the nonlinear discontinuity as the grid is refined. It is proved that there is convergence in the discrete approximation of linearized output functionals even for Dirac initial perturbations and pointwise convergence almost everywhere for the solution of the adjoint discrete equations. In particular, the adjoint approximation converges to the correct uniform value in the region in which characteristics propagate into the discontinuity. Moreover, it is shown that the results of [M. Giles and S. Ulbrich, SIAM J. Numer. Anal., 48 (2010), pp. 882–904] and the present paper hold also for quite general nonlinear initial data which contain multiple shocks and for which shocks form at a later time and/or merge. |
| first_indexed | 2024-03-06T20:34:34Z |
| format | Journal article |
| id | oxford-uuid:322eb67d-e78d-4a73-bbc5-d5fa9a6af999 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T20:34:34Z |
| publishDate | 2010 |
| publisher | SIAM |
| record_format | dspace |
| spelling | oxford-uuid:322eb67d-e78d-4a73-bbc5-d5fa9a6af9992022-03-26T13:12:31ZConvergence of linearized and adjoint approximations for discontinuous solutions of conservation laws. Part 2: adjoint approximations and extensionsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:322eb67d-e78d-4a73-bbc5-d5fa9a6af999Mathematical Institute - ePrintsSIAM2010Giles, MUlbrich, SThis paper continues the convergence analysis in [M. Giles and S. Ulbrich, SIAM J. Numer. Anal., 48 (2010), pp. 882–904] of discrete approximations to the linearized and adjoint equations arising from an unsteady one-dimensional hyperbolic equation with a convex flux function. We consider a simple modified Lax–Friedrichs discretization on a uniform grid, and a key point is that the numerical smoothing increases the number of points across the nonlinear discontinuity as the grid is refined. It is proved that there is convergence in the discrete approximation of linearized output functionals even for Dirac initial perturbations and pointwise convergence almost everywhere for the solution of the adjoint discrete equations. In particular, the adjoint approximation converges to the correct uniform value in the region in which characteristics propagate into the discontinuity. Moreover, it is shown that the results of [M. Giles and S. Ulbrich, SIAM J. Numer. Anal., 48 (2010), pp. 882–904] and the present paper hold also for quite general nonlinear initial data which contain multiple shocks and for which shocks form at a later time and/or merge. |
| spellingShingle | Giles, M Ulbrich, S Convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws. Part 2: adjoint approximations and extensions |
| title | Convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws. Part 2: adjoint approximations and extensions |
| title_full | Convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws. Part 2: adjoint approximations and extensions |
| title_fullStr | Convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws. Part 2: adjoint approximations and extensions |
| title_full_unstemmed | Convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws. Part 2: adjoint approximations and extensions |
| title_short | Convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws. Part 2: adjoint approximations and extensions |
| title_sort | convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws part 2 adjoint approximations and extensions |
| work_keys_str_mv | AT gilesm convergenceoflinearizedandadjointapproximationsfordiscontinuoussolutionsofconservationlawspart2adjointapproximationsandextensions AT ulbrichs convergenceoflinearizedandadjointapproximationsfordiscontinuoussolutionsofconservationlawspart2adjointapproximationsandextensions |