Journal of Computational Physics 289 (2015) 149–168 Contents lists available at ScienceDirect Journal of Computational Physics www.elsevier.com/locate/jcp A numerical solver for high dimensional transient Fokker–Planck equation in modeling polymeric fluids Yifei Sun ∗ , Mrinal Kumar 1 Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611-6250, USA a r t i c l e i n f o a b s t r a c t Article history: Received 8 September 2014 Received in revised form 8 January 2015 Accepted 18 February 2015 Available online 25 February 2015 Keywords: Fokker–Planck equation Polymeric fluids Tensor decomposition Chebyshev spectral method Curse of dimensionality Numerical partial differential equation In this paper, a tensor decomposition approach combined with Chebyshev spectral differentiation is presented to solve the high dimensional transient Fokker–Planck equations (FPE) arising in the simulation of polymeric fluids via multi-bead-spring (MBS) model. Generalizing the authors’ previous work on the stationary FPE, the transient solution is obtained in a single CANDECOMP/PARAFAC decomposition (CPD) form for all times via the alternating least squares algorithm. This is accomplished by treating the temporal dimension in the same manner as all other spatial dimensions, thereby decoupling it from them. As a result, the transient solution is obtained without resorting to expensive time stepping schemes. A new, relaxed approach for imposing the vanishing boundary conditions is proposed, improving the quality of the approximation. The asymptotic behavior of the temporal basis functions is studied. The proposed solver scales very well with the dimensionality of the MBS model. Numerical results for systems up to 14 dimensional state space are successfully obtained on a regular personal computer and compared with the corresponding matrix Riccati differential equation (for linear models) or Monte Carlo simulations (for nonlinear models).  2015 Elsevier Inc. All rights reserved. 1. Introduction In the study of polymeric fluids, the multi-bead-spring (MBS) model is commonly employed by rheologists to represent coarse-grained molecular configurations. This model is a chain of N beads connected by N − 1 springs, where the beads represent interaction points with the solvent and springs capture the local stiffness property depending on local stretch- ing [1]: see Fig. 1. For the special case of N = 2, it is often referred to as the dumbbell model. The configuration of an MBS chain can be specified by the connector vectors q i = r i+1 − r i , i = 1, ..., N − 1, where r i is the position vector of the beads with local dimensionality d equal to 1, 2 or 3. A typical flow characterizing the polymeric fluid is the shear flow with the following linear velocity field: v(x) = K x, K = β  0 1 0 0 0 0 0 0 0  , (1) where β is the shear rate. * Corresponding author. Graduate Research Assistant. E-mail addresses: yfsun@ufl.edu (Y. Sun), mrinalkumar@ufl.edu (M. Kumar). 1 Assistant Professor. http://dx.doi.org/10.1016/j.jcp.2015.02.026 0021-9991/ 2015 Elsevier Inc. All rights reserved.