Commun. Comput. Phys., 11 (2012), pp. 1144-1168. |
Improvement on Spherical Symmetry in Two-Dimensional Cylindrical Coordinates for a Class of Control Volume Lagrangian Schemes Juan Cheng ^{1}, Chi-Wang Shu ^{2*} 1 Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China.2 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. Received 3 July 2010; Accepted (in revised version) 13 December 2010 Available online 29 November 2011 doi:10.4208/cicp.030710.131210s Abstract In [14], Maire developed a class of cell-centered Lagrangian schemes for solving Euler equations of compressible gas dynamics in cylindrical coordinates. These schemes use a node-based discretization of the numerical fluxes. The control volume version has several distinguished properties, including the conservation of mass, momentum and total energy and compatibility with the geometric conservation law (GCL). However it also has a limitation in that it cannot preserve spherical symmetry for one-dimensional spherical flow. An alternative is also given to use the first order area-weighted approach which can ensure spherical symmetry, at the price of sacrificing conservation of momentum. In this paper, we apply the methodology proposed in our recent work [8] to the first order control volume scheme of Maire in [14] to obtain the spherical symmetry property. The modified scheme can preserve one-dimensional spherical symmetry in a two-dimensional cylindrical geometry when computed on an equal-angle-zoned initial grid, and meanwhile it maintains its original good properties such as conservation and GCL. Several two-dimensional numerical examples in cylindrical coordinates are presented to demonstrate the good performance of the scheme in terms of symmetry, non-oscillation and robustness properties. AMS subject classifications: 65M06, 76M20Notice: Undefined variable: pac in /var/www/html/issue/abstract/readabs.php on line 164 Key words: Control volume Lagrangian scheme, spherical symmetry preservation, conservative, cell-centered, compressible flow, cylindrical coordinates. *Corresponding author. Email: cheng_juan@iapcm.ac.cn (J. Cheng), shu@dam.brown.edu (C.-W. Shu) |