In this work, the model of Doi-Edwards with independant alignment approximation describing the dynamics of polymer melts is simulated. The main aim of this work is the analysis of some new simulation techniques operating on the Fokker-Planck equation related to that model. For this purpose we consider the kinetic theory description of the Doi-Edwards model, implemented in the 2D and 3D cases under shear and elongational flows. The Fokker Planck equation which governs the evolution of the distribution function involves two variables: the tube orientation (described by a unit vector defining the unit surface in 3D and the unit circle in 2D) and the coordinate that locates the segment tube on the molecular chain, taking values in the unit interval. To separate both variables during the problem resolution we make use of the Alternating Direction Implicit method (ADI) which allows reducing the computation time and efforts. A model reduction technique is also proposed and analyzed. It consists of considering an optimal representation basis which is constructed during the problem resolution. Thus, a reduced number of approximation functions, now defined in the whole domain, are enough to describe the solution evolution during the entire time interval considered in the simulation, with significant CPU time savings.
Modeling the Transport of Nonthermal Particles in Flares Using Fokker–Planck Kinetic Theory