Efficient algorithms for constraining orientation tensors in Galerkin methods for the Fokker–Planck equation

This paper deals with the problem of positivity preservation in numerical algorithms for simulating fiber suspension flows. In contrast to fiber orientation models based on the Advani–Tucker evolution equations for even-order orientation tensors, the probability distribution function of fiber orientation is approximated using the Galerkin discretization of the Fokker–Planck equation with Fourier basis functions or spherical harmonics. This procedure leads to a natural generalization of orientation tensor models replacing ad hoc closure approximations by Galerkin equations for the fine-scale components. After introducing an operator splitting approach to solving the discretized Fokker–Planck equation, we present conditions and correction techniques that guarantee physically correct distribution functions. As the reader will see, the derivation of these conditions is independent of the space dimension and their applicability is not limited to the simulation of fiber suspensions.