Applied/ACMS/absS15: Difference between revisions

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(Created page with "= ACMS Abstracts: Spring 2015 = === Irene Kyza (U Dundee) === ''Semiclassical regularization for ill-posed classical flows: microlocal coarse-graining beyond Wigner measures...")
 
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=== Irene Kyza (U Dundee) ===
=== Irene Kyza (U Dundee) ===


''Semiclassical regularization for ill-posed classical flows: microlocal coarse-graining beyond Wigner measures''
''Adaptivity and blowup detection for semilinear evolution convection-diffusion equations based on a posteriori error control''


Wigner measures (WMs) have been successfully used as a parameter-free tool to provide homogenised descriptions of wave problems. Notable applications are the efficient simulation of large linear wave fields, and the painless resolution of linear caustics. However, their applicability to non-linear problems has been very limited.
We discuss recent results on the a posteriori error control and adaptivity for an evolution semilinear convection-diffusion model problem with possible blowup in finite time. This belongs to the broad class of partial differential equations describing e.g., tumor growth,chemotaxis and cell modelling. In particular, we derive a posteriori error estimates that are conditional (estimates which are valid under conditions of a posteriori ­type) for an interior penalty discontinuous Galerkin (dG) implicit-explicit (IMEX) method using a continuation argument. Compared to a previous work, the obtained conditions are more localised and allow the efficient error control near the blowup time. Utilising the conditional a posteriori estimator we are able to propose an adaptive algorithm that appears to perform satisfactorily. In particular, it leads to good approximation of the blowup time and of the exact solution close to the blowup. Numerical experiments illustrate and complement our theoretical results. This is joint work with A. Cangiani, E.H. Georgoulis, and S. Metcalfe from the University of Leicester.
 
In this talk we discuss the role of smoothness of the underlying flow as a limiting factor in the applicability of WMs. Non-smooth flows are ill-posed for measures, and new phenomena are possible in that regime. For example, single wavepackets may be "split" cleanly into several new wavepackets. We introduce a modification of the WM approach, and show that it can capture successfully some of these new phenomena. These results include joint work with T. Paul, I. Kyza and Th. Katsaounis.
 
The main idea behind this regularised scheme can be used to  setup a unifying framework for several different approaches developed in the last few years. Some ideas about the extension of this framework to non-linear problems are also discussed.

Revision as of 05:24, 20 January 2015

ACMS Abstracts: Spring 2015

Irene Kyza (U Dundee)

Adaptivity and blowup detection for semilinear evolution convection-diffusion equations based on a posteriori error control

We discuss recent results on the a posteriori error control and adaptivity for an evolution semilinear convection-diffusion model problem with possible blowup in finite time. This belongs to the broad class of partial differential equations describing e.g., tumor growth,chemotaxis and cell modelling. In particular, we derive a posteriori error estimates that are conditional (estimates which are valid under conditions of a posteriori ­type) for an interior penalty discontinuous Galerkin (dG) implicit-explicit (IMEX) method using a continuation argument. Compared to a previous work, the obtained conditions are more localised and allow the efficient error control near the blowup time. Utilising the conditional a posteriori estimator we are able to propose an adaptive algorithm that appears to perform satisfactorily. In particular, it leads to good approximation of the blowup time and of the exact solution close to the blowup. Numerical experiments illustrate and complement our theoretical results. This is joint work with A. Cangiani, E.H. Georgoulis, and S. Metcalfe from the University of Leicester.

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