Reduced-Order Variational Deterministic-Particle-Based Scheme for Fokker-Planck Equations in Microscopic Polymer Dynamics

Abstract

This study proposes an acceleration technique for the computational challenges in extending the variational deterministic-particle-based scheme (VDS) [Bao et al., Journal of Computational Physics 522 (2025) 113589] to 3D complex fluid simulations with multi-bead polymers. While the original VDS effectively captures configuration space dynamics for 2D dumbbell polymers, its direct extensions reveal critical scalability limitations. The growing configuration space dimensionality necessitates prohibitively large particle ensembles to maintain distributional accuracy, so its quadratic computational cost scaling impedes practical applications. In this paper, we develop a model reduction framework integrating proper orthogonal decomposition (POD) to speed up the computation of the VDS for microscopic Fokker-Planck equations. Numerical validation using bead-spring chain models in simple shear flow demonstrates that the computational efficiency of the reduced model increases systematically with molecular complexity. The reduced-order model introduces about $6\%$ relative error in predicting the dynamics while requiring only about $6\%$ of the original computational time for $4$-bead chain polymers, where the relative numerical error of the reference dynamics is about $5\% \sim 10\%$, and the degrees of freedom can be reduced significantly to about $0.1\%$ of the original model, which means the low-dimensional structure is found by POD. This establishes a practical pathway for multiscale and complex fluid simulations.

Figures (12)

• Figure 1: Polymer stress $\boldsymbol{\tau}$ vs. evolution time $t$ in the reference dynamics. Parameters: $2$-bead polymer molecules without flow, particle number $P = 1000$, time step size $\delta _{t} = 0.001$.
• Figure 1: Various numerical error $\textup{err} _{\textup{time}}$ δ _t $$ vs. time step size $\delta _{t}$ in the reference dynamics. The left part shows the results from explicit schemes. Parameters: $4$-bead (homogeneous bonds) polymer molecules with simple shear flow, particle number $P = 1000$, reference time step size $\delta _{t} ^{0} = 5\times 10 ^{-4}$. The right part shows the results from implicit schemes. Parameters: the same as the left part.
• Figure 2: The bond of representative particles $\mathbf{q} _{1:}$ in the reference dynamics. From left to right, the plots correspond to evolution time $t = 0, 1.5, 3, 6 \textup{s}$. The bonds are illustrated in 3D space (yellow) and 2D projections on $x _{1} x _{2}$ (blue), $x _{2} x _{3}$ (red), $x _{3} x _{1}$ (green) planes. Parameters: $2$-bead polymer molecules without flow, particle number $P = 1000$, time step size $\delta _{t} = 0.001$.
• Figure 2: Various numerical error $\textup{err} _{\textup{poly-num}}$ P $$ vs. number of representative particles $P$ in the reference dynamics. Parameters: $4$-bead (homogeneous bonds) polymer molecules with simple shear flow, reference particle number $P ^{0} = 5000$, time step size $\delta _{t} = 0.001$, explicit scheme.
• Figure 3: Polymer stress $\boldsymbol{\tau}$ vs. evolution time $t$ in the reference dynamics. Parameters: $4$-bead (homogeneous bonds) polymer molecules with simple shear flow, particle number $P = 1000$, time step size $\delta _{t} = 0.001$.