Entropy Satisfying Discontinuous Galerkin Methods for Fokker-planck Equations, with Applications to the Finitely Extensible Nonlinear Elastic Dumbbell Model

Computation of Fokker-Planck equations with satisfying long time behavior is important in many applications and difficult in resolving solution structures induced by non-standard forces. Entropy satisfying conservative methods are proven to be powerful to ensure both equilibrium preserving and mass conservation properties at the discrete level. Following [H. Liu and H. Yu, SIAM Journal on Numerical Analysis 2012, 50(3), 1207-1239], we present entropy satisfying discontinuous Galerkin methods 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 methods satisfy two desired properties: mass conservation and entropy satisfying in the sense that these schemes are shown to satisfy discrete entropy inequalities for the quadratic entropy. These ensure that the schemes are entropy satisfying and preserve the equilibrium solutions. It is also proved the convergence of numerical solutions to the equilibrium solution as time becomes large. Zero-flux at the boundary is naturally incorporated and boundary behavior is resolved sharply. A positive numerical approximation is obtained with the same accuracy as the numerical solution through a reconstruction at the final time. Both one and two-dimensional numerical results are provided to demonstrate the good qualities of the schemes, as well as effects of some canonical homogeneous flows.