A Runge–Kutta discontinuous Galerkin method for viscous flow equations Hongwei Liu, Kun Xu * Mathematics Department, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong 10000, China Received 14 October 2005; received in revised form 18 October 2006; accepted 13 November 2006 Available online 12 January 2007 Abstract This paper presents a Runge–Kutta discontinuous Galerkin (RKDG) method for viscous flow computation. The con- struction of the RKDG method is based on a gas-kinetic formulation, which not only couples the convective and dissipa- tive terms together, but also includes both discontinuous and continuous representation in the flux evaluation at a cell interface through a simple hybrid gas distribution function. Due to the intrinsic connection between the gas-kinetic BGK model and the Navier–Stokes equations, the Navier–Stokes flux is automatically obtained by the present method. Numerical examples for both one dimensional (1D) and two dimensional (2D) compressible viscous flows are presented to demonstrate the accuracy and shock capturing capability of the current RKDG method. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Discontinuous Galerkin method; Gas-kinetic scheme; Viscous flow simulations 1. Introduction In the past decades, both the finite volume (FV) and the discontinuous Galerkin (DG) finite element meth- ods have been successfully developed for the compressible flow simulations. Most FV schemes use piecewise constant representation for the flow variables and employ the reconstruction techniques to obtain high accu- racy. A higher-order scheme usually has a larger stencil than a lower-order scheme, which makes it difficult to be applied on unstructured mesh or complicated geometry. For the DG method high-order accuracy is obtained by means of high-order approximation within each element, where more information is stored for each element during the computation. The compactness of the DG method allows it to deal with unstructured mesh or complicated geometry easily. Now the DG method has served as a high-order method for a broad class of engineering problems [6]. For viscous flow problems, many successful DG methods have been proposed in the literature, such as those by Bassi and Rebay [1], Cockburn and Shu [11], Baumann and Oden [14], and many others. In [5],a 0021-9991/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2006.11.014 * Corresponding author. Tel.: +852 2358 7433; fax: +852 2358 1643. E-mail addresses: hwliu@ust.hk (H. Liu), makxu@ust.hk (K. Xu). Journal of Computational Physics 224 (2007) 1223–1242 www.elsevier.com/locate/jcp