How to adaptively resolve evolutionary singularities in differential equations with symmetry

C J Budd, J F Williams

Research output: Contribution to journalArticle

16 Citations (Scopus)

Abstract

Many time-dependent partial differential equations have solutions which evolve to have features with small length scales. Examples are blow-up singularities and interfaces. To compute such features accurately it is essential to use some form of adaptive method which resolves fine length and time scales without being prohibitively expensive to implement. In this paper we will describe an r-adaptive method (based on moving mesh partial differential equations) which moves mesh points into regions where the solution is developing singular behaviour. The method exploits natural symmetries which are often present in partial differential equations describing physical phenomena. These symmetries give an insight into the scalings (of solution, space and time) associated with a developing singularity, and guide the adaptive procedure. In this paper the theory behind these methods will be developed and then applied to a number of physical problems which have (blow-up type) singularities linked to symmetries of the underlying PDEs. The paper is meant to be a practical guide towards solving such problems adaptively and contains an example of a Matlab code for resolving the singular behaviour of the semi-linear heat equation.
Original languageEnglish
Pages (from-to)217-236
Number of pages20
JournalJournal of Engineering Mathematics
Volume66
Issue number1-3
DOIs
Publication statusPublished - Mar 2010

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Partial differential equations
Resolve
Differential equations
Partial differential equation
Adaptive Method
Singularity
Differential equation
Symmetry
Length Scale
Blow-up
Moving Mesh
Semilinear Heat Equation
Adaptive Procedure
MATLAB
Time Scales
Mesh
Scaling
Form
Hot Temperature

Cite this

How to adaptively resolve evolutionary singularities in differential equations with symmetry. / Budd, C J; Williams, J F.

In: Journal of Engineering Mathematics, Vol. 66, No. 1-3, 03.2010, p. 217-236.

Research output: Contribution to journalArticle

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