Fast Jacobi Spectral Methods and Closure Approximations for the Homogeneous FENE Model of Complex Fluids

The Finitely Extensible Nonlinear Elastic (FENE) dumbbell model is a widely used mathematical model for complex fluids. Direct simulation of the FENE Fokker–Planck equation is computationally challenging due to high dimensionality and singularity of its potential. In this paper, we develop two fast Jacobi-Spherical Harmonic spectral methods for the spatially homogeneous FENE Fokker–Planck equation. These methods effectively resolve the singularity near the boundary by combining properly designed Jacobi polynomials with a weighted variational formulation. A semi-implicit backward differentiation formula of second-order (BDF2) is employed for time marching, and its energy stability is rigorously proved. The resulting linear algebraic system possesses a sparse structure and can be efficiently solved. Numerical results verify the spectral convergence and efficiency of the direct spectral solvers, establishing them as a reliable tool for generating reference solutions for challenging benchmark problems. Furthermore, to achieve an optimal trade-off between accuracy and efficiency, we compare several closure approximation models, including the industry workhorse Peterlin approximation (FENE-P), the quasi-equilibrium approximation (FENE-QE), and a novel neural network implementation for FENE-QE proposed in this paper (FENE-QE-NN). Numerical experiments in extensional and shear flows demonstrate the superior accuracy and efficiency of the proposed methods compared to traditional approaches.

💡 Research Summary

The paper addresses the computationally demanding problem of solving the spatially homogeneous Fokker–Planck equation associated with the finitely extensible nonlinear elastic (FENE) dumbbell model in three configuration dimensions. The authors develop two fast spectral solvers that combine Jacobi polynomials in the radial direction with real spherical harmonics for the angular variables, forming a Jacobi‑Spherical Harmonic (JSH) Galerkin discretization.

To handle the logarithmic singularity of the FENE potential at the unit‑ball boundary, they introduce a weighted transformation f(q,t) = (1‑|q|²)^s h(q,t) with 1 < s ≤ b/2, where b is the extensibility parameter. This yields a weighted weak formulation in the Sobolev space H¹_s(Ω) and a natural weighted inner product (·,·)_s. The radial coordinate r∈