Numerical stability of four positive (semi-)definite reformulations for viscoelastic fluid models in benchmark flows

In this paper we study the numerical stability of four positive (semi-)definite formulations for viscoelastic fluid flows: the LCR formulation of Fattal and Kupferman (2004), the Cholesky-log formulation of Vaithianathan and Collins (2003), the b-symmetric formulation of Balci et al. (2011) and the CDT formulation of Carrozza et al. (2019). The numerical stability is tested by evaluating the critical time step in four different flow problems: the flow around a cylinder confined between two walls, planar extrudate swell, an axisymmetric swell problem and the extension of the axisymmetric swell problem to three dimensions. Although LCR is a good performer throughout, the CDT formulation turns out to be the most stable/robust one. The numerical stability of the Cholesky-log and the b-symmetric formulation turns out to be much lower in at least one benchmark. This supports our hypothesis that a vector-like formulation increases numerical stability at least as much as the LCR formulation and that positive definiteness is not the main contribution to numerical stability.

Automated summary

This study compares the numerical stability of four positive (semi-)definite formulations for viscoelastic fluid flows, with the CDT formulation being identified as the most stable. The Cholesky-log and b-symmetric formulations show lower stability in at least one benchmark, supporting the hypothesis that a vector-like formulation can increase numerical stability as much as the LCR formulation. Positive definiteness is not the main contributing factor to numerical stability.