Technical note An improved low diffusion E-CUSP upwind scheme Ge-Cheng Zha ⇑ , Yiqing Shen, Baoyuan Wang Dept. of Mechanical and Aerospace Engineering, University of Miami, Coral Gables, FL 33124, United States article info Article history: Received 24 July 2010 Received in revised form 23 December 2010 Accepted 21 March 2011 Available online 2 April 2011 Keywords: Riemann solver Low diffusion abstract An improved low diffusion E-CUSP (LDE) scheme is presented. The E-CUSP scheme can capture crisp shock profile and exact contact surface. Several numerical cases are presented to demonstrate the accu- racy and robustness of the new scheme. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction With the application of computational fluid dynamics becoming more and more popular, the demand on high accuracy and high efficiency CFD solutions also becomes stronger to satisfy the needs of the broad engineering problems. To achieve efficiency, accuracy and simplicity, many efforts have been made to develop upwind schemes only using scalar dissipation instead of matrix dissipation such as that of the Roe’s flux difference splitting (FDS) scheme [1]. The examples include AUSM family schemes of Liou [2–6], the Van Leer–Hänel scheme [7], Edwards’s LDFSS schemes [8,9], Jameson’s CUSP schemes and limiters [10–12], and the E-CUSP schemes developed by Zha et al. [13–16], etc. Zha and Hu suggested an E-CUSP schemes, which has low diffu- sion and can capture crisp shock wave profiles and exact contact discontinuities [16]. The scheme is consistent with the characteris- tic directions due to the nature of E-CUSP scheme. The scheme shows the highest stability for two shock tube tests problems com- pared with several other popularly used upwind schemes for the explicit Euler time marching scheme. The scheme also works well when extended to multi-dimensions [16]. However, the E-CUSP scheme of Zha–Hu may generate temperature oscillation near the computation boundary, in particular when the mesh is skewed. Zha was able to remove the temperature oscillation by introducing the total enthalpy in the smooth factor for the energy equation [17,18]. However, the scheme loses the capability to capture the exact contact surface due to the modification. The purpose of this paper is to present an improved low diffu- sion and efficient E-CUSP upwind scheme that is able to capture crisp shock profile and exact contact surface, and is smooth for multi-dimensional flow calculations. This paper modifies the Zha–Hu E-CUSP scheme [16] by using the Mach number splitting of Edwards’s LDFSS schemes [8] for the convective flux. The scheme is shown to be accurate, robust and efficient by the cases tested in this paper. 2. The numerical scheme 2.1. Governing equations To describe the new scheme, we will begin with the quasi-1D Euler equations in Cartesian coordinates for inviscid flow: @ t U þ @ x E  H ¼ 0 ð1Þ where U ¼ SQ ; Q ¼ q qu qe 0 @ 1 A ; E ¼ SF, F ¼ qu qu 2 þ p ðqe þ pÞu 0 B @ 1 C A; H ¼ dS dx 0 p 0 0 B @ 1 C A ð2Þ In above equations, q is the density, u is the velocity, p is the static pressure, e is the total energy per unit mass and S is the cross sec- tional area of the 1D duct. The following equation is also employed to relate pressure with the conservative variables: p ¼ðc  1Þ qe  1 2 qu 2   ð3Þ where c is the specific heat ratio with the value of 1.4 for ideal gas. The finite volume method with the explicit Euler temporal inte- gration is used to discretize the governing equations. It yields the following formulation at cell i: 0045-7930/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compfluid.2011.03.012 ⇑ Corresponding author. Tel.: +1 305 284 3328; fax: +1 305 284 2580. E-mail address: gzha@miami.edu (G.-C. Zha). Computers & Fluids 48 (2011) 214–220 Contents lists available at ScienceDirect Computers & Fluids journal homepage: www.elsevier.com/locate/compfluid