Semi-Lagrangian advection

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... 

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... 

McDonald A (1987) Accuracy of multiply-stream, semi-lagrangian advective schemes II. Mon Weather Rev 115 1446-11450... 

J. Behrens, A. Iske, and S. Pohn (2001) Effective node adaption for grid-free semi-Lagrangian advection. Discrete Modelling and Discrete Algorithms in Continuum Mechanics, T. Sonar and I. Thomas (eds.). Logos, Berlin, 110-119. 

A. Iske and M. Kaser (2004) Conservative semi-Lagrangian advection on adaptive unstructured meshes. To appear in Numer. Meth. Part. Diff. Eq. 

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). 

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. 

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

This paper presents a mathematical model and numerical analysis of momentum transport and heat transfer of polymer melt flow in a standard cooling extruder. The finite element method is used to solve the three-dimensional Navier-Stokes equations based on a moving barrel formulation a semi-Lagrangian approach based on an operator-splitting technique is used to solve the heat transfer advection-diffusion equation. A periodic boundary condition is applied to model fully developed flow. The effects of polymer properties on melt flow behavior, and the additional effects of considering heat transfer, are presented. 

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