Energy dissipative characteristic schemes for the diffusive Oldroyd‐B viscoelastic fluid

Abstract

Summary In this paper, we propose new energy dissipative characteristic numerical methods for the approximation of diffusive Oldroyd‐B equations that are based either on the finite element or finite difference discretization. We prove energy stability of both schemes and illustrate their behavior on a series of numerical experiments. Using both the diffusive model and the logarithmic transformation of the elastic stress, we are able to obtain methods that converge as mesh parameter is refined. Copyright © 2015 John Wiley & Sons, Ltd.

Energy-stable linear schemes for polymer–solvent phase field models

“…In this Subsection we present yet another possibility to discretize system (15). In order to save computational costs we split the computation into three different substeps.…”

“…This discretization is mass conserving for φ. The proof of the energy dissipation property for the fully discrete schemes can be done in an analogous way to that for the semi-discrete schemes, see Lukáčová-Medvid'ová et al [15]. 220…”

“…Similarly, topological changes of the interface structure are automatically described correctly. The physics (and therefore also the mathematics and numerics) becomes more involved if one component -or both 15 -is a macromolecular compound. In this case, the large molecular relaxation time gives rise to a dynamic coupling between intra-molecular processes and the unmixing on experimentally relevant time scales, with interesting new phenomena, for which the term "viscoelastic phase separation" [8] has been coined.…”

“…In order to describe the dynamics of a complex polymer-solvent mixture, the Cahn-Hilliard equations for the phase field evolution are coupled with the 35 Oldroyd-B equations, which consist of the momentum equation for the velocity field, the continuity equation, and the rheological equation for time evolution of the elastic stress tensor. We note in passing that there is quite a large number of analytical as well as numerical results available in the literature for the Oldroyd-B system, see, e. g., [12,13,14,15]. The main challenge in this field is to obtain 40 a stable approximate numerical solution for large Weissenberg numbers.…”

“…In the present work we consider the non-critical regime of Weissenberg numbers. Applying the techniques from [13,15], a further dynamics of the polymer-solvent mixture without any hydrodynamic effects. Section 3 is devoted to problem-suited numerical methods for both models.…”