On the numerical solution of one-dimensional PDEs using adaptive methods based on equidistribution

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Abstract

Numerical experiments are described that illustrate some important features of the performance of moving mesh methods for solving one-dimensional partial differential equations (PDEs). The particular method considered here is an adaptive finite difference method based on the equidistribution of a monitor function and it is one of the moving mesh methods proposed by W. Huang, Y. Ren, and R. D. Russell (1994, SIAM J. Numer. Anal.31 709). We show how the accuracy of the computations is strongly dependent on the choice of monitor function, and we present a monitor function that yields an optimal rate of convergence. Motivated by efficiency considerations for problems in two or more space dimensions, we demonstrate a robust and efficient algorithm in which the mesh equations are uncoupled from the physical PDE. The accuracy and efficiency of the various formulations of the algorithm are considered and a novel automatic time-step control mechanism is integrated into the scheme.
LanguageEnglish
Pages372-392
Number of pages20
JournalJournal of Computational Physics
Volume167
Issue number2
DOIs
Publication statusPublished - 2001

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partial differential equations
Partial differential equations
monitors
mesh
Finite difference method
formulations
Experiments

Keywords

  • computational physics
  • differential equations

Cite this

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abstract = "Numerical experiments are described that illustrate some important features of the performance of moving mesh methods for solving one-dimensional partial differential equations (PDEs). The particular method considered here is an adaptive finite difference method based on the equidistribution of a monitor function and it is one of the moving mesh methods proposed by W. Huang, Y. Ren, and R. D. Russell (1994, SIAM J. Numer. Anal.31 709). We show how the accuracy of the computations is strongly dependent on the choice of monitor function, and we present a monitor function that yields an optimal rate of convergence. Motivated by efficiency considerations for problems in two or more space dimensions, we demonstrate a robust and efficient algorithm in which the mesh equations are uncoupled from the physical PDE. The accuracy and efficiency of the various formulations of the algorithm are considered and a novel automatic time-step control mechanism is integrated into the scheme.",
keywords = "computational physics, differential equations",
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T1 - On the numerical solution of one-dimensional PDEs using adaptive methods based on equidistribution

AU - Beckett, G.

AU - Ramage, A.

AU - Sloan, D.M.

AU - Mackenzie, J.A.

PY - 2001

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AB - Numerical experiments are described that illustrate some important features of the performance of moving mesh methods for solving one-dimensional partial differential equations (PDEs). The particular method considered here is an adaptive finite difference method based on the equidistribution of a monitor function and it is one of the moving mesh methods proposed by W. Huang, Y. Ren, and R. D. Russell (1994, SIAM J. Numer. Anal.31 709). We show how the accuracy of the computations is strongly dependent on the choice of monitor function, and we present a monitor function that yields an optimal rate of convergence. Motivated by efficiency considerations for problems in two or more space dimensions, we demonstrate a robust and efficient algorithm in which the mesh equations are uncoupled from the physical PDE. The accuracy and efficiency of the various formulations of the algorithm are considered and a novel automatic time-step control mechanism is integrated into the scheme.

KW - computational physics

KW - differential equations

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