## 1. Introduction

Semi-Lagrangian advection schemes have become favoured methods in numerical weather prediction (NWP) (Williamson, 2007) because they are more accurate and efficient than traditional Eulerian schemes in terms of large time steps. Staniforth and Cote (1991) provided an excellent review of semi-Lagrangian methods for atmospheric models. Recently, most semi-Lagrangian methods have focused on mass conservation (Xiao *et al.*, 2002; Lauritzen *et al.*, 2006; Cotter *et al.*, 2007; Zerroukat *et al.*, 2007; Juang, 2008; Kaas, 2008; Juang and Hong, 2010).

The first crucial issue in a semi-Lagrangian scheme is to efficiently and accurately determine the trajectories in either backward schemes or forward schemes. Traditional backward schemes compute trajectories with an iterative algorithm. Several other economical schemes have been proposed using a truncated Taylor series approximation in backward schemes (McGregor, 1993; Hortal, 2002; Lauritzen *et al.*, 2006) and forward schemes (Nair *et al.*, 2003). This iterative procedure is a computationally expensive component of the semi-Lagrangian scheme and is worse in the polar zones where special treatment and approximations are required (Nair and Machenhauer, 2002; Zerroukat *et al.*, 2009). There exists a non-interpolating semi-Lagrangian scheme (Ritchie, 1986). It decomposes the trajectory vector into the integer part and the fractional part. The fractional advection is done by the Eulerian upstream scheme. Lin and Rood (1996) extended it to a flux-form scheme for mass conservation.

Many recent semi-Lagrangian methods use conservative reconstruction and remapping schemes that are finite volume methods where the conservation variable is an integrated quantity over a certain finite control volume, such as the piecewise parabolic method (PPM) (Colella and Woodward, 1984), parabolic spline method (PSM) (Zerroukat *et al.*, 2006), piecewise quartic method (PQM) (White and Adcroft, 2008) and quartic spline method (QSM) (Zerroukat *et al.*, 2010). Filters and/or limiters preserving monotonicity are used (Lin, 2004; Zerroukat *et al.*, 2006; Colella and Sekora, 2008).

In extending 1D schemes to higher dimensions, one choice is to get trajectories into higher-dimension space and use cascade remapping to split the high dimensions into two one-dimensional steps, because fully two-dimensional reconstruction is not easily monotone and positive. Cascade schemes are a little complicated because they apply one-dimensional remapping, first from the regular Eulerian cells to the intermediate grid cells along the Eulerian longitudes or latitudes, and second from the intermediate grid cells to the departure cells along the Lagrangian latitudes or longitudes (which is a curve), especially in the SLICE scheme (Figure 1 of Zerroukat *et al.*, 2007). Another choice is operator splitting, which splits the advection process into a combination of operators in each of the two coordinate directions.

Several locally mass-conserved semi-Lagrangian schemes have been proposed. Kaas (2008) modifies the upstream interpolation coefficient area weights that transfer information from arrival Eulerian grid points to departure semi-Lagrangian points, which ensures that the total mass given off by a given Eulerian grid point to all the surrounding departure points is equal to the cell area represented by this grid point. The forecast densities in the arrival Eulerian grid points, including the effects of divergence, are equal to the modified upstream interpolated values divided by the cell area represented by the arrival Eulerian grid point.

Juang (2007, 2008) proposed a non-iteration dimen-sional-split semi-Lagrangian (NDSL) scheme, which we will describe in section 2. A two-dimensional NDSL will be presented in section 3. In section 4, we followed Kaas' test cases to compare the NDSL with other semi-Lagrangian schemes from the literature. Discussion and conclusions are given in section 5.