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On the solution of the viscous Burgers' equation with nonlinear viscosity
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
2004 (English)Report (Other academic)
Abstract [en]

The viscous Burgers' equation with nonlinear viscosity is considered. The equation is written as a quasilinear parabolic equation in divergence form, and existence of a weak solution is shown. The proof is based on Galerkin approximations which converges in a suitable Banach space. Finally, the Cole-Hopf transformation is used to derive an analytical solution in the case when the viscosity is constant. This solution turns out to be very ill-conditioned for numerical evaluations. The solution can be rewritten with the Poisson summation formula. Comparisons to a finite difference solution are done.

Place, publisher, year, edition, pages
Linköping: Linköpings universitet , 2004. , 18 p.
Series
LiTH-MAT-R, ISSN 0348-2960
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-22770ISRN: LITH-MAT-R-2004-07Local ID: 2097OAI: oai:DiVA.org:liu-22770DiVA: diva2:243083
Available from: 2009-10-07 Created: 2009-10-07 Last updated: 2013-11-04
In thesis
1. Reconstruction of velocity data using adjoint optimization
Open this publication in new window or tab >>Reconstruction of velocity data using adjoint optimization
2004 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

In many application areas there is a growing interest in data assimilation or data reconstruction. Data assimilation is a process for integrating observed or measured data into a physical model. The problem originates from a vast array of different topics: traditionally in metereological and oceanographic modelling, and recently from non-invasive medical measurement devices such as magnetic resonance imaging. The measured data may contain inaccurancies and random noise, given with low spatial and/or temporal resolution.

This thesis presents a method for solving reconstruction problems in fluid dynamics using optimal control theory. The problem considered here includes a known partial differential equation and some spatially and temporarily sparsely distributed data with an unknown initial state. From a given velocity field uδ, a flow field u is determined which satisfies a given system of partial differential equations and minimizes || u - u*|| L2. The function u(x,t) is known at the boundary and the initial condition u0(x) is used as design variable. The optimization problem is solved using adjoint formulation.

Place, publisher, year, edition, pages
Linköping: Linköpings universitet, 2004. 8 p.
Series
Linköping Studies in Science and Technology. Thesis, ISSN 0280-7971 ; 1096
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-23162 (URN)2566 (Local ID)91-7373-969-3 (ISBN)2566 (Archive number)2566 (OAI)
Presentation
2004-05-25, Glashuset, Hus B, Linköpings universitet, Linköpings, 13:15 (Swedish)
Opponent
Available from: 2009-10-07 Created: 2009-10-07 Last updated: 2013-11-04

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Lundvall, JohanWeinerfelt, Per

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