JID:APNUM AID:3285 /FLA [m3G; v1.226; Prn:8/12/2017; 8:35] P.1(1-14) Applied Numerical Mathematics ••• (••••) •••–••• Contents lists available at ScienceDirect Applied Numerical Mathematics www.elsevier.com/locate/apnum The upwind hybrid difference methods for a convection diffusion equation Youngmok Jeon ∗,1 , Mai Lan Tran Department of Mathematics, Ajou University, Suwon, 16499, Republic of Korea a r t i c l e i n f o a b s t r a c t Article history: Available online xxxx Keywords: Convection diffusion equations Hybrid difference method Penalty method Upwind method We propose the upwind hybrid difference method and its penalized version for the convection dominated diffusion equation. The hybrid difference method is composed of two types of approximations: one is the finite difference approximation of PDEs within cells (cell FD) and the other is the interface finite difference (interface FD) on edges of cells. The interface finite difference is derived from continuity of normal fluxes. The penalty method is obtained by adding small diffusion in the interface FD. The penalty term makes it possible to reduce severe numerical oscillations in the upwind hybrid difference solutions. The penalty parameter is designed to be some power of the grid size. A complete stability is provided. Convergence estimates seems to be conservative according to our numerical experiments. To exposit convergence property and controllability of numerical oscillations several numerical tests are provided.  2017 IMACS. Published by Elsevier B.V. All rights reserved. 1. Introduction Let us consider the convection dominated diffusion equation. −ǫ u + β ·∇u = f on , (1.1a) u = 0 on Ŵ = ∂ , (1.1b) where 0 < ǫ ≪ 1 and ‖β ‖ = 1, and the domain  is simply connected. Many physical and engineering phenomena including fluid mechanics and transport problems are governed by convec- tion dominated diffusion equations in which the magnitudes of the diffusion coefficients are much smaller than those of the convection coefficients. The solutions of those problems contain boundary and interior layers, and standard numerical methods usually yield numerical solutions with spurious and non-physical oscillations on a relatively coarse mesh, which come mainly from instability of numerical schemes. To overcome these difficulties, many numerical methods have been developed. Among the finite element methods proposed for (1.1), the upwind methods are the earliest and simple stabilized numerical schemes [1,13,26]. The upwind finite element methods are obtained by mimicking the upwind finite difference in the finite element framework. These methods are stable but very diffusive and of first order accuracy. [6]. An important progress was made by Brooks and Hughes [6], who introduced the SUPG (streamline upwind Petrov Galerkin) method. The * Corresponding author. E-mail addresses: yjeon@ajou.ac.kr (Y. Jeon), mailan@ajou.ac.kr (M.L. Tran). 1 This author was supported by NRF 2015R1D1A1A09057935. https://doi.org/10.1016/j.apnum.2017.12.002 0168-9274/ 2017 IMACS. Published by Elsevier B.V. All rights reserved.