Evaluating the performance of the Inexact-Newton-Krylov scheme using globalization and forcing terms for non-Newtonian flows

Abstract. Non-Newtonian fluids are widely spread in industry. Examples are polymer processing, paint, food production or drilling muds. The dependence of the viscosity on the shear rate adds nonlinearity to the governing equations which complicates solving the transient, incompressible Navier-Stokes equation. Here, we use a semi-discrete stabilized finite element formulation for the governing equation. Often Newton-type algorithms are used to solve the resulting system of nonlinear equations at each time step. Those algorithms can converge rapidly from a good initial guess. However, it may appear that they are too expensive, since exact solutions of the linearized system are required for each iteration step. Therefore, the Inexact Newton-Krylov method (INK) is used to solve the linearized system of the Newton-scheme, reducing the computational effort. Hereby, the balance between the accuracy and the amount of effort per iteration is described by a tolerance, the so-called forcing term. Globalization strategies, like backtracking or trust region methods, are used to enhance the robustness of the INK algorithm. In this study the effects of a globalization strategy and several forcing terms of the Inexact-Newton-Krylov are evaluated. As a globalization strategy a backtracking method is applied. We compare four different forcing terms to verify which one has the best convergence. To do so, we simulate a Bingham fluid of a benchmark cavity and Taylor-Couette flow, both in three-dimensions, and analyze nonlinear and linear convergence effects. We compare the number of linear iterations and CPU time. Results are analyzed and discussed aiming to establish guidelines for an effective INK utilization in practice. Keywords: Inexact Newton-Krylov, backtracking, non-Newtonian fluids, forcing terms