Semi-Lagrangian

Leip A, Lammel G (2004) Indicators for persistence and long-range transport potential as derived from multicompartment chemistrytransport modelling. Environ Poll 128 205-221 Lin SJ, Rood RB (1996) Multidimensional flux form semi-Lagrangian transport. Mon Wea Rev 124 2046-2068... 

Lin S-J, Rood RB (1996) Multidimensional flux-form semi-Lagrangian transport schemes. Mon Weather Rev 124 2046-2070... 

Nair R, Machenhauer B (2002) The mass-conservative cell-integrated semi-Lagrangian advection scheme on the sphere. Mon Weather Rev 130(3) 649-667... 

Calculate tendencies due to semi-Lagrangian advection scheme... 

McGregor JL (1997) Semi-Lagrangian advection on a cubic gnomonic projection of the sphere. In Lin C, Laprise R, Ritchie H (eds) Numerical methods in atmospheric and oceanic modelling the Andre J. Robert Memorial (Companion volume to atmosphere-ocean). Canadian Meteorological and Oceanographic Society, Ottawa, Canada, pp 153-169... 

The test setup works for schemes on flux-form but does not apply to semi-Lagrangian schemes (interpolation of equal numbers) or schemes on advective form (difference of equal numbers). 

It would be good to find an optimal advection scheme for integrated ACT-HIRLAM, however, estimations of ACT and NWP modellers can be different, because they can use different requirements/criteria for the best scheme for NWP and ACTM. At least it is reasonable to analyze and compare different schemes used in the HIRLAM community (for NWP and ACT including the semi-Lagrangian, CISL, Bott, Easter, Chlond, Walcek, Galperin and Kaas). 

Kaas E (2008) A simple and efficient locally mass conserving semi-Lagrangian transport scheme. Tellus 60A 305-320... 

Sensitivity Studies on 1969 Trajectories with the Expanded Model. Based on the semi-Lagrangian formulation of the photochemical/diffusion model, the computed endpoint composition of the air masses depends on initial conditions, flux from the ground along the trajectories, and reaction rates. For our tests we concentrate on El Monte data because much of the polluted air there comes from somewhere else. This is believed to be a more severe test of the model than that at Huntington Park. The initial conditions are based on measurements insofar as possible. The principal initial values for the 1030 trajectory are as follows for 0730 PST (given in parts per hundred million) ... 

Besides the mathematical improvements, the atmospheric model has been adapted to a semi-Lagrangian formulation. By following selected air masses, we avoid commitments of large quantities of memory and the incursions of artificial diffusion errors. Most important, we do not end up with stacks of computer printout that relate to regions where there are no measurements. Also, predictive calculations will become more useful, but our present levels of resources and sophistication demand that effort be concentrated on verification. Only in this way can the confidence be built that is needed for applying modeling techniques to implementation planning. 

High-order convection/advection schemes are widely used in meteorological applications solving h3rperbolic equations. For example, in European weather forecast models the explicit non-flux-based modified methods of characteristics have been very popular as they are very fast. Typical examples of this type of schemes are the semi-Lagrangian advection schemes of Bates and McDonald [9], McDonald [129] and McDonald [130]. These methods have an unrestricted time step advantage, but also an important disadvantage that they are not strictly conservative due to their non-flux-based formulation. 

Bates JR, McDonald A (1982) Multiply-Upstream, Semi-Lagrangian Advective Schemes Analysis and Application to a Multi-Level Primitive Equation Model. Mon Wea Rev 110 1831-1842... 

McDonald A (1984) Accuracy of Multiply-Stream, Semi-Lagrangian Advective Schemes. Mon Wea Rev 112 1267-1275... 

Staniforth, A., and J. Cote, Semi-Lagrangian integration schemes for middle atmosphere models—A review. Mon Weath Rev 119, 2206, 1991. 

Sniolarkiewicz, P. K. and Rash, P. J. (1991). Monotone advection on the sphere an Eulerian versus semi-Lagrangian approach./. Atmos. Sci. 48, 793-810. 

Mesoscale models have predominately utilized the finite-difference and (for advection), the semi-Lagrangian approaches. A few groups have applied the finite-element method, but its additional computational cost has limited its use. The spectral method, which is most valuable for models without boundaries and for simulation without sharp spatial gradients, has not generally been used since mesoscale models have lateral boundaries. 

Benoit, R., Desgagne, M., Pellerin, P, Pellerin, S., Desjardins, S., and Chattier, Y. (1997). The Canadian MC2 A semi-Lagrangian, semi-implicit wide band atmospheric model suited for five-scale process studies and simulation, Monfh/y Weather Rev. 125, 2382-2415. 

Semi-Lagrangian method Another numerical time integration approach based on the Lagrangian method, which calcrrlates the properties of fluid following the fluid parcels. To avoid the distortion of parcel trajectories, new fltrid parcels ate selected at legrtlarly distributed grid points at each time step. 

We present the basic idea of semi-Lagrangian method using a simple one-dimensional conservation (or advec-tion) equation... 

FIGURE 1 Three-time-level grid structure for the semi-Lagrangian scheme applied to the one-dimensional advection equation (38). See text for details. 

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