Consistent splitting schemes for incompressible viscoelastic flow problems

Viscoelastic fluids are highly challenging from the rheological standpoint, and their discretization demands robust, efficient numerical solvers. Simulating viscoelastic flows requires combining Navier–Stokes system with a dynamic tensorial equation, increasing mathematical computational demands. Hence, fractional-step methods decoupling calculation of flow quantities an attractive option. In consistent schemes, splitting equations is derived continuous level, so that neither mass nor momentum balance sacrificed. Thus, no corrections or velocity projections needed, resulting in fewer algorithmic steps than other classical approaches. Moreover, consistency guarantees absence both boundary layers errors, enabling high-order accuracy space time. This article introduces first for incompressible fluids, which arbitrary constitutive laws allowed. We present formulation algorithm, along various examples testing accuracy. First-, second- third-order backward-differentiation schemes numerically tested, optimal convergence confirmed several spatial temporal discretizations. Furthermore, good stability verified benchmark problems, including steady time-dependent solutions.