A Simple 2nd-order Physically Bounded Convective Scheme

Abstract

In this paper, we focus on the transient convection problems of which physically bounds are already known. In tradition, the high-resolutions (HR) schemes that use some parameters to measure the local monotonicity in the predicted scalar field, including the well-known TVD-type and NVD-type schemes, are widely used. Recently, Qin Qian, et al., [1] proposed a new approach which use the upwind approximation as the 'unbounded indicator". During the computation, the actual scheme switches between QUICK scheme and 1st-order upwind (FUD) scheme, according to whether the indicator locates in the range of physical bounds or not. However, the new approach does not apply to unstructured grids as the far-upwind point for QUICK scheme is not easy to determine. Considering the easy implementation of central differencing (CD) scheme on structured and unstructured meshes, we propose a new convective scheme following the idea of [1], but using CD scheme as the high-order base scheme. The new scheme is extremely simple, but with 2nd-order accuracy. And best of all, the scheme can be applied to arbitrary grids. Testing is carried out on the 1D (transport of a top-hat pulse) problem on a uniform grid. Further testing on the problem of advection of a square-shaped scalar field on structured and unstructured grids are also done. Finally, we compare the solution obtained by FUD, Gamma, MUSCL and the proposed scheme and find that the new scheme performs well for all the test cases considered.