Free-energy-dissipative schemes for the Oldroyd-B model

TL;DR

This paper analyzes the stability of various numerical schemes for the Oldroyd-B model of viscoelastic fluids, focusing on free energy dissipation properties and comparing different discretizations, including log-formulation methods.

Contribution

It provides a theoretical analysis of free energy dissipation in numerical schemes for the Oldroyd-B model, especially for log-formulation discretizations, explaining their improved stability.

Findings

Log-formulation schemes better preserve free energy dissipation.

Stability conditions depend on specific discretization assumptions.

Analysis explains observed numerical stability improvements.

Abstract

In this article, we analyze the stability of various numerical schemes for differential models of viscoelastic fluids. More precisely, we consider the prototypical Oldroyd-B model, for which a free energy dissipation holds, and we show under which assumptions such a dissipation is also satisfied for the numerical scheme. Among the numerical schemes we analyze, we consider some discretizations based on the log-formulation of the Oldroyd-B system proposed by Fattal and Kupferman, which have been reported to be numerically more stable than discretizations of the usual formulation in some benchmark problems. Our analysis gives some tracks to understand these numerical observations.