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Assessment of a high-order finite difference upwind scheme for the simulation of convection-diffusion problems

Ferreira, V.G. and Kurokawa, F.A. and Queiroz, R.A.B. and Kaibara, M.K. and Oishi, C.M. and Cuminato, J.A. and Castelo, A. and Tomé, M.F. and McKee, S. (2009) Assessment of a high-order finite difference upwind scheme for the simulation of convection-diffusion problems. International Journal of Numerical Methods in Fluids, 60 (1). pp. 1-26. ISSN 0271-2091

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Abstract

This article deals with the study of the development and application of the high-order upwind ADBQUICKEST scheme, an adaptative bounded version of the QUICKEST for unsteady problems (Commun. Numer. Meth. Engng 2007; 23:419-445), employing both linear and nonlinear convection term discretization. This scheme is applicable to a wide range of computational fluid dynamics problems, where transport phenomena are of special importance. In particular, the performance of the scheme is assessed through an extensive numerical simulation study of advection-diffusion problems. The scheme, implemented in the context of finite difference methodology, combines a good approximation of shocks (or discontinuities) with a good approximation of the smooth parts of the solutions. In order to assess the performance of the scheme, seven problems are solved, namely (a) advection of scalars; (b) non-linear viscous Burgers equation; (c) Euler equations of gas dynamics; (d) Newtonian flow in a channel; (e) axisymmetric Newtonian jet flow; (f) axisymmetric non-Newtonian (generalized Newtonian) flow in a pipe; and (g) collapse of a fluid column. The numerical experiments clearly show that the scheme provides more consistent solutions than those found in the literature. From the study, the flexibility and robustness of the ADBQUICKEST scheme is confirmed by demonstrating its capability to solve a variety of linear and nonlinear problems with and without discontinuous solutions.