A conservative scheme for the Fokker-Planck equation with applications to viscoelastic polymeric fluids

We propose a conservative scheme for a high-dimensional Fokker-Planck equation that arises in the dynamics of infinitely extensible polymer molecules. This leads to a challenging problem of unbounded domain. Our scheme combines the Lagrange-Galerkin method and the Hermite spectral method together with a space splitting approach. We prove that the scheme preserves the discrete mass. Combining it with a stabilized Lagrange-Galerkin method for the Navier-Stokes equations, we further propose a multiscale scheme for the simulation of some viscoelastic polymeric fluids. Several numerical experiments are presented to illustrate the performance of the schemes, and to confirm the conservation of mass at the discrete level.