On the approximation of the Fokker-Planck equation of the finitely extensible nonlinear elastic dumbbell model. I: A new weighted formulation and an optimal spectral-Galerkin algorithm in two dimensions. (English) Zbl 1250.82033
Summary: We propose a new weighted weak formulation for the Fokker-Planck equation of the finitely extensible nonlinear elastic dumbbell model and prove its well-posedness in weighted Sobolev spaces. We also propose simple and efficient semi-implicit time-discretization schemes which are unconditionally stable, i.e., the step size of time marching does not depend on the number of the bases used in the configurational space. We then restrict ourselves to the two-dimensional case and construct two Fourier-Jacobi spectral-Galerkin algorithms which enjoy the following properties: (i) they are unconditionally stable, spectrally accurate, and of optimal computational complexity; (ii) they conserve the volume and provide an accurate approximation to higher-order moments of the distribution function; and (iii) they can be easily extended to coupled nonhomogeneous systems. Numerical results are presented to show how to choose a proper weight to get the best numerical results of the distribution function and the polymer stress.
MSC:
82C31 | Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics |
82-08 | Computational methods (statistical mechanics) (MSC2010) |
82D60 | Statistical mechanics of polymers |
35K20 | Initial-boundary value problems for second-order parabolic equations |
65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
35Q84 | Fokker-Planck equations |
35Q30 | Navier-Stokes equations |