A Fourth-Order Compact Finite Difference Scheme for Solving Unsteady Convection-Diffusion Equations

Abstract

Convection-diffusion equations are widely used for modeling and simulations of various complex phenomena in science and engineering (Hundsdorfer & Verwer, 2003; Morton, 1996). Since for most application problems it is impossible to solve convection-diffusion equations analytically, efficient numerical algorithms are becoming increasingly important to numerical simulations involving convection-diffusion equations. Recently a great deal of efforts have been devoted to developing high-order compact schemes, which utilize only the grid nodes directly adjacent to the central node. In (Noye & Tan, 1989), Noye and Tan derived a class of high-order implicit schemes for solving the one-dimensional unsteady convection-diffusion equations. Thismethod is very stable and accurate (third-order in space and second-order in time). In (Gupta et al., 1984), a fourth-order finite difference scheme for a steady convection-diffusion equation with variable coefficients was proposed. The scheme is defined on a single square cell of size 2Δx over a nine-point stencil. In (Rigal, 1994), Rigal provided an extensive analysis of the properties of a class of twoand three-level second-order difference schemes which have been proposed in (Rigal, 1989; 1990). In (Spotz & Carey, 2001), the two-dimensional HOC (High Order Compact) scheme proposed in (Gupta et al., 1984) was extended to solve unsteady one-dimensional convection-diffusion equations with variable coefficients and two-dimensional diffusion equations. This method was further extended by Kalita et al. in (Kalita et al., 2002) to a class of HOC schemes with weighted time discretization, and successfully used to solve unsteady two-dimensional convection-diffusion equations. In (Karaa & Zhang, 2004), Karaa and Zhang proposed a novel high-order alternating direction implicit method, based on the technique in (Zhang et al., 2002), for solving unsteady two-dimensional convection-diffusion problems. This new method is second-order in time and fourth-order in space, and is computationally efficient. In (Tian & Dai, 2007), Tian and Dai proposed a class of high-order compact exponential finite difference methods for solving oneand two-dimensional steady-state convection-diffusion problems. This method is nonoscillatory, fourth-order in space, and easy to implement. Some more recent high-order ADI methods for unsteady convection-diffusion equations can be found in (Tian & Ge, 2007; You, 2006). 4