On the reduction of kinetic theory models related to finitely extensible dumbbells

Stochastic simulation for finitely extensible non-linear elastic (FENE) dumbbells has been successfully applied (see the review paper of Keunings [R. Keunings, Micro–macro methods for the multiscale simulation viscoelastic flow using molecular models of kinetic theory, in: D.M. Binding, K. Walters (Eds.), Rheology Reviews, British Society of Rheology, 2004, pp. 67–98] and the references therein). The main difficulty in these simulations is related to the high number of realizations required for describing accurately the microstructural state due to Brownian effects.The discretisation of the Fokker–Planck equation with a mesh support (finite elements, finite differences, finite volumes, spectral techniques, ……) allows to go beyond the difficulty related to Brownian effects. However, kinetic theory models involve physical and conformation spaces. Thus, the molecular distribution depends on time, space as well as on the molecular orientation and extension (conformation coordinates). In this form the resulting Fokker–Planck equation is defined in a space of dimension 7.In the reduction technique proposed in this paper, a reduced approximation basis is constructed. The new shape functions are defined in the whole domain in an appropriate manner. Thus, the number of degrees of freedom involved in the solution of the Fokker–Planck equation is significantly reduced. The construction of those new approximation functions is done with an ‘a priori’ approach, which combines a basis reduction (using the Karhunen–Loève decomposition) with a basis enrichment based on the use of some Krylov subspaces. This numerical technique is applied for solving the FENE model of viscoelastic flows.