On adaptive timestepping for weakly instationary solutions of hyperbolic conservation laws via adjoint error control

We study a recent timestep‐adaptation technique for hyperbolic conservation laws. The key tool is a space–time splitting of adjoint error representations for target functionals due to Süli (An Introduction to Recent Developments in Theory and Numerics for Conservation Laws. Lecture Notes in Computat...

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Veröffentlicht in:	International journal for numerical methods in biomedical engineering Jg. 26; H. 6; S. 790 - 806
Hauptverfasser:	Steiner, Christina, Noelle, Sebastian
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Chichester, UK John Wiley & Sons, Ltd 01.06.2010 Wiley
Schlagworte:	adaptive timestepping adjoint error control Adjoints Classical transport Computation Conservation laws Errors Exact sciences and technology Fluid dynamics Fundamental areas of phenomenology (including applications) Galerkin methods General theory hyperbolic conservation laws Mathematical analysis Mathematical models Physics Representations Statistical physics, thermodynamics, and nonlinear dynamical systems Transport processes weakly instationary solutions adjoint error control Transport equation Steady flow Hyperbolic equations Duality Orthogonality Conservation laws Galerkin-Petrov method Finite element method Direct problem Dies hyperbolic conservation laws weakly instationary solutions adaptive timestepping Modelling
ISSN:	2040-7939, 2040-7947, 2040-7947
Online-Zugang:	Volltext
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Zusammenfassung:	We study a recent timestep‐adaptation technique for hyperbolic conservation laws. The key tool is a space–time splitting of adjoint error representations for target functionals due to Süli (An Introduction to Recent Developments in Theory and Numerics for Conservation Laws. Lecture Notes in Computational Science and Engineering. Springer: Berlin, 1998; 123–194) and Hartmann (A posteriori Fehlerschätzung und adaptive Schrittweiten‐ und Ortsgittersteuerung bei Galerkin‐Verfahren für die Wärmeleitungsgleichung. Diplomarbeit, Institut für Angewandte Mathematik, Universität Heidelberg, 1998). It provides an efficient choice of timesteps for implicit computations of weakly instationary flows. The timestep will be very large in regions of stationary flow and become small when a perturbation enters the flow field. Besides using adjoint techniques that are already well established, we also add a new ingredient that simplifies the computation of the dual problem. Owing to Galerkin orthogonality, the dual solution φ does not enter the error representation as such. Instead, the relevant term is the difference of the dual solution and its projection to the finite element space, φ−φh . We can show that it is therefore sufficient to compute the spatial gradient of the dual solution, w=∇φ . This gradient satisfies a conservation law instead of a transport equation, and it can therefore be computed with the same algorithm as the forward problem, and in the same finite element space. We demonstrate the capabilities of the approach for a weakly instationary test problem for scalar conservation laws. Copyright © 2008 John Wiley & Sons, Ltd.
Bibliographie:	ArticleID:CNM1183 istex:EBA8A8E1274463FF3807E8C210E893F0837634EE ark:/67375/WNG-CHGQ5RJH-N DFG - No. SFB 401 ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 23 ObjectType-Article-2 ObjectType-Feature-1
ISSN:	2040-7939 2040-7947 2040-7947
DOI:	10.1002/cnm.1183