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Mathematical Problems in Engineering
Volume 2014 (2014), Article ID 176752, 20 pages
Research Article

A Third Order Accurate Cellwise Relaxation Implicit Discontinuous Galerkin Scheme for Unstructured Hybrid Meshes

1Department of Aerospace Engineering, Tohoku University, 6-6-01 Aramaki-Aza-Aoba, Aoba-ku, Sendai 980-8579, Japan
2Institute of Aeronautical Technology, Japan Aerospace Exploration Agency, 7-44-1 Jindaiji-Higashi, Chofu, Tokyo 182-8522, Japan

Received 29 December 2013; Accepted 27 May 2014; Published 8 July 2014

Academic Editor: Yonghong Wu

Copyright © 2014 Hiroyuki Asada et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A third order accurate cellwise relaxation implicit Discontinuous Galerkin (DG) scheme for RANS simulations using unstructured hybrid meshes is presented. A scalar parabolic equation is first examined to clarify what is really important in construction of implicit matrix to keep its diagonal dominance for the third and fourth order cellwise relaxation implicit DG schemes. In addition, discussions are given to approximated construction of implicit matrix for reducing computational cost. Then, the third order accurate cellwise relaxation implicit DG scheme for RANS simulations is successfully developed utilizing the expertise learned in the study of solving the parabolic equation. Superior spatial accuracy of the third order accurate cellwise relaxation implicit DG scheme for RANS simulations, while retaining reasonable convergence properties, is demonstrated for typical aerospace applications.