An Entropy Satisfying Conservative Method for the Fokker–Planck Equation of the Finitely Extensible Nonlinear Elastic Dumbbell Model

Abstract

In this paper, we propose an entropy satisfying conservative method to solve the Fokker–Planck equation of the finitely extensible nonlinear elastic dumbbell model for polymers, subject to homogeneous fluids. Both semidiscrete and fully discrete schemes satisfy all three desired properties—(i) mass conservation, (ii) positivity preserving, and (iii) entropy satisfying—in the sense that these schemes satisfy discrete entropy inequalities for both the physical entropy and the quadratic entropy. These ensure that the computed solution is a probability density and the schemes are entropy stable and preserve the equilibrium solutions. We also prove convergence of the numerical solution to the equilibrium solution as time becomes large. Zero flux at boundary is naturally incorporated, and boundary behavior is resolved sharply. Both one- and two-dimensional numerical results are provided to demonstrate the good qualities of the scheme and the effects of some canonical homogeneous flows.