Moving mesh generation using the parabolic Monge–Ampère equation

Chris J Budd, J F Williams

Research output: Contribution to journalArticle

40 Citations (Scopus)

Abstract

This article considers a new method for generating a moving mesh which is suitable for the numerical solution of partial differential equations (PDEs) in several spatial dimensions. The mesh is obtained by taking the gradient of a (scalar) mesh potential function which satisfies an appropriate nonlinear parabolic partial differential equation. This method gives a new technique for performing r-adaptivity based on ideas from optimal transportation combined with the equidistribution principle applied to a (time-varying) scalar monitor function (used successfully in moving mesh methods in one-dimension). Detailed analysis of this new method is presented in which the convergence, regularity, and stability of the mesh is studied. Additionally, this new method is shown to be straightforward to program and implement, requiring the solution of only one simple scalar time-dependent equation in arbitrary dimension, with adaptivity along the boundaries handled automatically. We discuss three preexisting methods in the context of this work. Examples are presented in which either the monitor function is prescribed in advance, or it is given by the solution of a partial differential equation.
Original languageEnglish
Pages (from-to)3438-3465
Number of pages28
JournalSIAM Journal on Scientific Computing
Volume31
Issue number5
Early online date3 Sep 2009
DOIs
Publication statusPublished - 2009

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Moving Mesh
Monge-Ampère Equation
Mesh generation
Mesh Generation
Partial differential equations
Parabolic Equation
Adaptivity
Scalar
Mesh
Monitor
Partial differential equation
Moving Mesh Method
Optimal Transportation
Equidistribution
Parabolic Partial Differential Equations
Potential Function
Nonlinear Partial Differential Equations
One Dimension
Time-varying
Regularity

Cite this

Moving mesh generation using the parabolic Monge–Ampère equation. / Budd, Chris J; Williams, J F.

In: SIAM Journal on Scientific Computing, Vol. 31, No. 5, 2009, p. 3438-3465.

Research output: Contribution to journalArticle

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