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

### Description

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

### Creation Information

Stinis, Panagiotis June 10, 2007.

## Who

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## What

### Description

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.

### Source

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

### Identifier

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• Report No.: LBNL--62851
• Grant Number: DE-AC02-05CH11231
• Office of Scientific & Technical Information Report Number: 929763

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## When

### Creation Date

• June 10, 2007

### Added to The UNT Digital Library

• Sept. 27, 2016, 1:39 a.m.

### Description Last Updated

• Sept. 22, 2017, 3:02 p.m.

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