Section: New Results
Positivity preserving schemes
Participants : Roland Becker, Daniela Capatina.
The stability and robustness with respect to physical parameters of numerical schemes for polymer flows is a challenging question. Indeed, most algorithms encounter serious convergence problems for large Weissenberg numbers. In the recent years, this issue has been asscociated to the discrete positive definiteness of the so-called conformation tensor. It seems therefore essential for the numerical simulations to employ positivity preserving schemes.
In order to develop such schemes, we have adopted the approach proposed by Lee and Xu  in the case of the quasi-linear Oldroyd-B model, based on the similarities between its constitutive equation and differential Riccati equations. We have applied it to a more general matrix equation; a typical example is the nonlinear Giesekus constitutive law. In agreement with our code (see Section 6.7 ), we have discretized it by a discontinuous Galerkin method combined with an upwinding of the transport terms, whereas the approach of Lee and Xu relies essentially on the characteristics method.
We have shown that a modification of Newton's method yields a monotone and positive scheme, under certain hypothesis, and we have applied this study to both Oldroyd-B and Giesekus polymer models. This allowed us to better understand and explain the better behaviour of the latter for large Weissenberg numbers. These results have been presented in  , .
We are working on several challenging questions such as the extension to other discretization methods, the improvement of the iterative method or the derivation of energy estimates for the coupled system.