Numerical methods for simulating dilute polymeric fluids - Ye, S (2018)
Navier--Stokes--Fokker--Planck systems are coupled systems of partial differential equations arising from statistical physics as mathematical models of dilute polymeric fluids. The solvent is assumed to be a viscous incompressible Newtonian fluid, whose evolution in time is modelled by the Navier--Stokes equations; the elastic effects exhibited by the dilute polymeric fluid are modelled by the elastic extra stress tensor, whose spatial divergence appears on the right-hand side of the Navier--Stokes momentum equation. In this thesis we investigate and implement several numerical methods for the computational simulation of dilute polymeric fluids. The Fokker--Planck equation featuring in the model is a high-dimensional transport-diffusion equation, whose numerical solution by conventional means is extremely challenging since standard numerical methods applied to this equation suffer from the curse of dimensionality. The key objective of the thesis is therefore to develop an efficient numerical approximation scheme for the model, where deterministic approximation techniques for the Fokker--Planck equation are replaced by a multilevel Monte Carlo method. Under suitable assumptions, we can prove exponential convergence in time of the solution to an equilibrium solution: the Maxwellian of the model. At the end of the thesis we perform a series of numerical experiments, including multi-bead simulations for polymer molecules modelled as bead-spring chains, with FENE-type (finitely extensible nonlinear elastic) spring potentials, to explore the practical performance of the proposed numerical method.