Semi-Lagrangian formulation

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. 

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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