Dimensional reduction as a tool for mesh refinement and trackingsingularities of PDEs

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We present a collection of algorithms which utilizedimensional reduction to perform mesh refinement and study possiblysingular solutions of time-dependent partial differential equations. Thealgorithms are inspired by constructions used in statistical mechanics toevaluate the properties of a system near a critical point. The firstalgorithm allows the accurate determination of the time of occurrence ofa possible singularity. The second algorithm is an adaptive meshrefinement scheme which can be used to approach efficiently the possiblesingularity. Finally, the third algorithm uses the second algorithm untilthe available resolution is exhausted (as we approach the possiblesingularity) and then switches to a dimensionally reduced model which,when accurate, ... continued below

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Stinis, Panagiotis June 10, 2007.

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We present a collection of algorithms which utilizedimensional reduction to perform mesh refinement and study possiblysingular solutions of time-dependent partial differential equations. Thealgorithms are inspired by constructions used in statistical mechanics toevaluate the properties of a system near a critical point. The firstalgorithm allows the accurate determination of the time of occurrence ofa possible singularity. The second algorithm is an adaptive meshrefinement scheme which can be used to approach efficiently the possiblesingularity. Finally, the third algorithm uses the second algorithm untilthe available resolution is exhausted (as we approach the possiblesingularity) and then switches to a dimensionally reduced model which,when accurate, can follow faithfully the solution beyond the time ofoccurrence of the purported singularity. An accurate dimensionallyreduced model should dissipate energy at the right rate. We construct twovariants of each algorithm. The first variant assumes that we have actualknowledge of the reduced model. The second variant assumes that we knowthe form of the reduced model, i.e., the terms appearing in the reducedmodel, but not necessarily their coefficients. In this case, we alsoprovide a way of determining the coefficients. We present numericalresults for the Burgers equation with zero and nonzero viscosity toillustrate the use of the algorithms.

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  • Journal Name: Journal of Computational Physics; Journal Volume: 0; Journal Issue: 0; Related Information: Journal Publication Date: 0

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  • Report No.: LBNL--62851
  • Grant Number: DE-AC02-05CH11231
  • Office of Scientific & Technical Information Report Number: 929763
  • Archival Resource Key: ark:/67531/metadc899222

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  • June 10, 2007

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  • Sept. 27, 2016, 1:39 a.m.

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  • Sept. 22, 2017, 3:02 p.m.

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Stinis, Panagiotis. Dimensional reduction as a tool for mesh refinement and trackingsingularities of PDEs, article, June 10, 2007; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc899222/: accessed September 23, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.