On the solution of the Fokker–Planck equation using a high-order reduced basis approximation

The numerical solution of the one-dimensional Fokker–Planck equation for describing the evolution of the configuration probability density function associated with kinetic theory models in polymer dynamics is presented. The finitely extensible non-linear elastic (FENE) model is considered and the spectral element discretisation is applied using an adaptive reduced basis technique. This technique facilitates a significant reduction in the number of degrees of freedom required in the approximation through the determination of an optimal basis using the singular value decomposition (SVD). The basis functions are constructed dynamically so that the numerical approximation is optimal in the current finite-dimensional subspace of the solution space. This is achieved through basis enrichment and projection. The reduced basis method is extended to the high-dimensional Fokker–Planck equation, using the d-dumbbell FENE model, by consideration of a high-dimensional singular value decomposition. Some numerical results are presented to demonstrate the efficiency of the numerical scheme.