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Consistency, accuracy and entropy behaviour of remeshed particle methods

Published online by Cambridge University Press:  31 July 2012

Lisl Weynans  and
Adrien Magni
Lisl Weynans
Affiliation:
Univ. Bordeaux, IMB, UMR 5251, 33400 Talence, France. lisl.weynans@math.u-bordeaux1.fr CNRS, IMB, UMR 5251, 33400 Talence, France INRIA, 33400 Talence, France
Adrien Magni
Affiliation:
Laboratoire Jean Kuntzmann, Université Joseph Fourier, 51 rue des Mathématiques, 38041 Grenoble Cedex 9, France; adrien.magni@imag.fr
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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.

Keywords

Particle methods with remeshinginterpolation kernelsconsistencytruncation errorentropy inequalitiestotal variationlimitersconvergence
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 1 , January 2013 , pp. 57 - 81
Copyright
© EDP Sciences, SMAI, 2012

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