Volume 6, Issue 2
A Numerical Study of Blowup in the Harmonic Map Heat Flow Using the MMPDE Moving Mesh Method

Ronald D. Haynes, Weizhang Huang & Paul A. Zegeling

Numer. Math. Theor. Meth. Appl., 6 (2013), pp. 364-383.

Published online: 2013-06

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  • Abstract

The numerical solution of the harmonic heat map flow problems with blowup in finite or infinite time is considered using an adaptive moving mesh method. A properly chosen monitor function is derived so that the moving mesh method can be used to simulate blowup and produce accurate blowup profiles which agree with formal asymptotic analysis. Moreover, the moving mesh method has finite time blowup when the underlying continuous problem does.  In situations where the continuous problem has infinite time blowup, the moving mesh method exhibits finite time blowup with a blowup time tending to infinity as the number of mesh points increases. The inadequacy  of a uniform mesh solution is clearly demonstrated.

  • Keywords

Heat flow, harmonic map, blowup, moving mesh method, finite difference.

  • AMS Subject Headings

35K91, 65M50, 65M06

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-6-364, author = {}, title = {A Numerical Study of Blowup in the Harmonic Map Heat Flow Using the MMPDE Moving Mesh Method}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2013}, volume = {6}, number = {2}, pages = {364--383}, abstract = {

The numerical solution of the harmonic heat map flow problems with blowup in finite or infinite time is considered using an adaptive moving mesh method. A properly chosen monitor function is derived so that the moving mesh method can be used to simulate blowup and produce accurate blowup profiles which agree with formal asymptotic analysis. Moreover, the moving mesh method has finite time blowup when the underlying continuous problem does.  In situations where the continuous problem has infinite time blowup, the moving mesh method exhibits finite time blowup with a blowup time tending to infinity as the number of mesh points increases. The inadequacy  of a uniform mesh solution is clearly demonstrated.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2013.1130nm}, url = {http://global-sci.org/intro/article_detail/nmtma/5909.html} }
TY - JOUR T1 - A Numerical Study of Blowup in the Harmonic Map Heat Flow Using the MMPDE Moving Mesh Method JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 364 EP - 383 PY - 2013 DA - 2013/06 SN - 6 DO - http://doi.org/10.4208/nmtma.2013.1130nm UR - https://global-sci.org/intro/article_detail/nmtma/5909.html KW - Heat flow, harmonic map, blowup, moving mesh method, finite difference. AB -

The numerical solution of the harmonic heat map flow problems with blowup in finite or infinite time is considered using an adaptive moving mesh method. A properly chosen monitor function is derived so that the moving mesh method can be used to simulate blowup and produce accurate blowup profiles which agree with formal asymptotic analysis. Moreover, the moving mesh method has finite time blowup when the underlying continuous problem does.  In situations where the continuous problem has infinite time blowup, the moving mesh method exhibits finite time blowup with a blowup time tending to infinity as the number of mesh points increases. The inadequacy  of a uniform mesh solution is clearly demonstrated.

Ronald D. Haynes, Weizhang Huang & Paul A. Zegeling. (2020). A Numerical Study of Blowup in the Harmonic Map Heat Flow Using the MMPDE Moving Mesh Method. Numerical Mathematics: Theory, Methods and Applications. 6 (2). 364-383. doi:10.4208/nmtma.2013.1130nm
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