High-order, low dispersive and low dissipative explicit schemes for multiple-scale and boundary problems Julien Berland, Christophe Bogey, Olivier Marsden, Christophe Bailly * Laboratoire de Me´canique des Fluides et d’Acoustique, UMR CNRS 5509, Ecole Centrale de Lyon, 69134 Ecully, France Received 17 May 2005; received in revised form 9 October 2006; accepted 12 October 2006 Available online 28 November 2006 Abstract Explicit high-order numerical schemes are proposed for the accurate computation of multiple-scale problems and for the implementation of boundary conditions. Specific high-order node-centered finite differences and selective filters remov- ing grid-to-grid oscillations are first designed for the discretization of the buffer region between a Dx-grid domain and 2Dx- grid domain. The coefficients of these matching schemes are chosen so that the maximum order of accuracy is reached. Non-centered finite differences and selective filters are then developed with the aim of accurately computing boundary con- ditions. They are constructed by minimizing the dispersion and the dissipation errors in the wave number space for waves down to four points per wavelength. The dispersion and dissipation properties of the matching and the boundary schemes are described in detail, and their accuracy limits are determined, to show that these schemes calculate accurately waves with at least five points per wavelength. Test problems, including linear convection, wall reflection and acoustic scattering around a cylinder, are finally solved to illustrate the accuracy of the schemes. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Multiple-scales problem; Boundary conditions; Explicit schemes; High order; Low dispersion; Low dissipation; Matching schemes; Non-centered schemes 1. Introduction Since the earliest stages in computational aeroacoustics (CAA), the need for highly accurate schemes has been recognized [1]. To meet the stringent accuracy requirements of CAA, low dispersive, low dissipative and large spectral bandwidth numerical methods have been designed by optimizing their dispersion and dis- sipation properties in the Fourier space for low wave numbers. Available optimized finite differences are for instance the explicit dispersion-relation-preserving scheme of Tam and Webb [2] or the compact schemes of Lele [3]. Bogey and Bailly [4] also recently developed explicit finite differences and selective filters accurate for waves down to four points per wavelength, allowing the direct computation of aerodynamic noise using 0021-9991/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2006.10.017 * Corresponding author. Fax: +33 4 72 18 91 43. E-mail addresses: julien.berland@ec-lyon.fr (J. Berland), christophe.bogey@ec-lyon.fr (C. Bogey), olivier.marsden@ec-lyon.fr (O. Marsden), christophe.bailly@ec-lyon.fr (C. Bailly). Journal of Computational Physics 224 (2007) 637–662 www.elsevier.com/locate/jcp