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

TL;DR

This paper introduces fast spectral methods for solving the FENE Fokker-Planck equation in complex fluids, addressing computational challenges and improving accuracy with novel closure models and neural network approximations.

Contribution

Develops two efficient Jacobi-spherical harmonic spectral methods for the FENE Fokker-Planck equation, and compares closure models including a new neural network approach.

Findings

Spectral convergence and efficiency of the proposed methods.

Superior accuracy of the new methods over traditional approaches.

Effective resolution of boundary singularity in the FENE model.

Abstract

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.