Consistency, accuracy and entropy behaviour of remeshed particle methods

Lisl Weynans; Adrien Magni

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2013)

  • Volume: 47, Issue: 1, page 57-81
  • ISSN: 0764-583X

Abstract

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In this paper we analyze the consistency, the accuracy and some entropy properties of particle methods with remeshing in the case of a scalar one-dimensional conservation law. As in [G.-H. Cottet and L. Weynans, C. R. Acad. Sci. Paris, Ser. I 343 (2006) 51–56] we re-write particle methods with remeshing in the finite-difference formalism. This allows us to prove the consistency of these methods, and accuracy properties related to the accuracy of interpolation kernels. Cottet and Magni devised recently in [G.-H. Cottet and A. Magni, C. R. Acad. Sci. Paris, Ser. I 347 (2009) 1367–1372] and [A. Magni and G.-H. Cottet, J. Comput. Phys. 231 (2012) 152–172] TVD remeshing schemes for particle methods. We extend these results to the nonlinear case with arbitrary velocity sign. We present numerical results obtained with these new TVD particle methods for the Euler equations in the case of the Sod shock tube. Then we prove that with these new TVD remeshing schemes the particle methods converge toward the entropy solution of the scalar conservation law.

How to cite

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Weynans, Lisl, and Magni, Adrien. "Consistency, accuracy and entropy behaviour of remeshed particle methods." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.1 (2013): 57-81. <http://eudml.org/doc/273119>.

@article{Weynans2013,
abstract = {In this paper we analyze the consistency, the accuracy and some entropy properties of particle methods with remeshing in the case of a scalar one-dimensional conservation law. As in [G.-H. Cottet and L. Weynans, C. R. Acad. Sci. Paris, Ser. I 343 (2006) 51–56] we re-write particle methods with remeshing in the finite-difference formalism. This allows us to prove the consistency of these methods, and accuracy properties related to the accuracy of interpolation kernels. Cottet and Magni devised recently in [G.-H. Cottet and A. Magni, C. R. Acad. Sci. Paris, Ser. I 347 (2009) 1367–1372] and [A. Magni and G.-H. Cottet, J. Comput. Phys. 231 (2012) 152–172] TVD remeshing schemes for particle methods. We extend these results to the nonlinear case with arbitrary velocity sign. We present numerical results obtained with these new TVD particle methods for the Euler equations in the case of the Sod shock tube. Then we prove that with these new TVD remeshing schemes the particle methods converge toward the entropy solution of the scalar conservation law.},
author = {Weynans, Lisl, Magni, Adrien},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {particle methods with remeshing; interpolation kernels; consistency; truncation error; entropy inequalities; total variation; limiters; convergence; Burgers equation; nonlinear scalar transport equation; infinite domain; finite difference scheme; total-variation diminishing-remeshing scheme; nonlinear conservation law; Euler equation},
language = {eng},
number = {1},
pages = {57-81},
publisher = {EDP-Sciences},
title = {Consistency, accuracy and entropy behaviour of remeshed particle methods},
url = {http://eudml.org/doc/273119},
volume = {47},
year = {2013},
}

TY - JOUR
AU - Weynans, Lisl
AU - Magni, Adrien
TI - Consistency, accuracy and entropy behaviour of remeshed particle methods
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 1
SP - 57
EP - 81
AB - In this paper we analyze the consistency, the accuracy and some entropy properties of particle methods with remeshing in the case of a scalar one-dimensional conservation law. As in [G.-H. Cottet and L. Weynans, C. R. Acad. Sci. Paris, Ser. I 343 (2006) 51–56] we re-write particle methods with remeshing in the finite-difference formalism. This allows us to prove the consistency of these methods, and accuracy properties related to the accuracy of interpolation kernels. Cottet and Magni devised recently in [G.-H. Cottet and A. Magni, C. R. Acad. Sci. Paris, Ser. I 347 (2009) 1367–1372] and [A. Magni and G.-H. Cottet, J. Comput. Phys. 231 (2012) 152–172] TVD remeshing schemes for particle methods. We extend these results to the nonlinear case with arbitrary velocity sign. We present numerical results obtained with these new TVD particle methods for the Euler equations in the case of the Sod shock tube. Then we prove that with these new TVD remeshing schemes the particle methods converge toward the entropy solution of the scalar conservation law.
LA - eng
KW - particle methods with remeshing; interpolation kernels; consistency; truncation error; entropy inequalities; total variation; limiters; convergence; Burgers equation; nonlinear scalar transport equation; infinite domain; finite difference scheme; total-variation diminishing-remeshing scheme; nonlinear conservation law; Euler equation
UR - http://eudml.org/doc/273119
ER -

References

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