The Use of TVD Limiters for Forward-in-Time Upstream-Biased Advection Schemes in Ocean Modeling

Abstract

This paper explores the use of the constant grid flux form forward-in-time upstream-biased advection schemes for the advection of temperature and salinity in ocean modeling. The constant grid flux form schemes are shown to be an improvement over the traditional central differencing commonly used in ocean models. In addition, nonoscillatory versions of the scheme, which employ flux limiters, are explored. The limiters are based on total variation diminishing concepts and are applied to higher-order (in space) versions of the constant grid flux form scheme. The constant grid flux form schemes are Crowley-type upstream-biased Eulerian advection schemes. They are mass conserving and possess small amplitude and phase errors. The flux limiters prevent the under- and overshooting associated with the numerical dispersion of the unlimited schemes. The limited schemes are easy to implement, efficient, and nonoscillatory. Of these schemes the third-order and fifth-order versions employing the PDM limiter are shown to give optimal results for the advection of scalars under a number of test cases. There is a trade-off between the greater accuracy of the fifth-order scheme and the requirement of a larger grid stencil associated with a greater computational cost. Higher-order nonoscillatory schemes can easily be developed for three-dimensional primitive equation modeling. A multidimensional version is presented that employs a modified time-splitting technique. It is shown to be suitable for three-dimensional primitive equation modeling.