SIAM J. SCI. COMPUT. c  2005 Society for Industrial and Applied Mathematics Vol. 27, No. 3, pp. 995–1013 A COMPARISON OF TROUBLED-CELL INDICATORS FOR RUNGE–KUTTA DISCONTINUOUS GALERKIN METHODS USING WEIGHTED ESSENTIALLY NONOSCILLATORY LIMITERS ∗ JIANXIAN QIU † AND CHI-WANG SHU ‡ Abstract. In [SIAM J. Sci. Comput., 26 (2005), pp. 907–929], we initiated the study of using WENO (weighted essentially nonoscillatory) methodology as limiters for the RKDG (Runge–Kutta discontinuous Galerkin) methods. The idea is to first identify “troubled cells,” namely, those cells where limiting might be needed, then to abandon all moments in those cells except the cell averages and reconstruct those moments from the information of neighboring cells using a WENO methodol- ogy. This technique works quite well in our one- and two-dimensional test problems [SIAM J. Sci. Comput., 26 (2005), pp. 907–929] and in the follow-up work where more compact Hermite WENO methodology is used in the troubled cells. In these works we used the classical minmod-type TVB (total variation bounded) limiters to identify the troubled cells; that is, whenever the minmod limiter attempts to change the slope, the cell is declared to be a troubled cell. This troubled-cell indicator has a TVB parameter M to tune and may identify more cells than necessary as troubled cells when M is not chosen adequately, making the method costlier than necessary. In this paper we system- atically investigate and compare a few different limiter strategies as troubled-cell indicators with an objective of obtaining the most efficient and reliable troubled-cell indicators to save computational cost. Key words. Runge–Kutta discontinuous Galerkin method, limiters, weighted essentially non- oscillatory finite volume scheme, high-order accuracy AMS subject classifications. 65M60, 65M99, 35L65 DOI. 10.1137/04061372X 1. Introduction. In [13], we initiated the study of using weighted essentially nonoscillatory (WENO) methodology as limiters for the Runge–Kutta discontinuous Galerkin (RKDG) methods. The idea is to first identify “troubled cells,” namely, those cells where limiting might be needed, then to abandon all moments in those cells except the cell averages and reconstruct those moments from the information of neighboring cells using a WENO methodology. This technique works quite well in our one- and two-dimensional test problems [13] and in the follow-up work [14] and [15], where more compact Hermite WENO methodology was used in the troubled cells. In these earlier works we adopted the classical minmod-type total variation bounded (TVB) limiters as in [5, 3, 7] to identify the troubled cells; that is, whenever the minmod limiter attempts to change the slope, the cell is declared to be a troubled cell. This troubled-cell indicator has a TVB parameter M to tune. When M is taken too small, more cells than necessary may be identified as troubled cells. Even though ∗ Received by the editors August 20, 2004; accepted for publication (in revised form) April 19, 2005; published electronically December 8, 2005. http://www.siam.org/journals/sisc/27-3/61372.html † Department of Mathematics, Nanjing University, Nanjing, Jiangsu 210093, People’s Republic of China, and Department of Mechanical Engineering, National University of Singapore, Singa- pore 119260 (jxqiu@nju.edu.cn). This author’s research was partially supported by NNSFC grant 10371118 and by NUS research project R-265-000-118-112. ‡ Division of Applied Mathematics, Brown University, Providence, RI 02912 (shu@dam.brown. edu). This author’s research was partially supported by the Chinese Academy of Sciences while the author was in residence at the University of Science and Technology of China (grant 2004-1-8) and at the Institute of Computational Mathematics and Scientific/Engineering Computing. Additional support was provided by ARO grant W911NF-04-1-0291, NSF grant DMS-0207451, and AFOSR grant F49620-02-1-0113. 995