Consistency, accuracy and entropy behaviour of remeshed particle methods

¹ Univ. Bordeaux, IMB, UMR 5251, 33400 Talence, France
lisl.weynans@math.u-bordeaux1.fr
² CNRS, IMB, UMR 5251, 33400 Talence, France
³ INRIA, 33400 Talence, France
⁴ Laboratoire Jean Kuntzmann, Université Joseph Fourier, 51 rue des Mathématiques, 38041 Grenoble Cedex 9, France
adrien.magni@imag.fr

Received: 12 October 2011
Revised: 5 April 2012

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.