Speaker
Description
The flow of a Newtonian incompressible viscous fluid is a fundamental problem in computational sciences and engineering. In this case, the governing equations are the incompressible Navier–Stokes equations. For decades, researches have focused their interest on devising numerical methods for the solution of this problem. The non-linearity of the incompressible Navier–Stokes equations requires one to employ robust and efficient solvers for either the Picard or the Newton linearization of the discretized equations, which results in a sequence of linear systems to be solved in order to obtain a numerical solution. For instationary problems, the importance of a robust and efficient linear solver is even more evident, as at each time step one is required to solve a sequence of linear systems.
In this talk, we consider the numerical integration of the instationary incompressible Navier–Stokes equations, when employing a Runge–Kutta method in time. The time discretization results in a non-linear system to be solved for the stages of the Runge–Kutta method at each time step. In order to find a numerical solution, we employ a Newton linearization of the non-linear problem, which is then discretized with suitable finite elements. The resulting linear systems present a saddle-point block structure, and can be very large and sparse in real-life applications. For this reason, in order to find a solution one requires the use of preconditioned iterative methods. We employ an augmented Lagrangian strategy, and apply the preconditioner in mixed precision arithmetic. Numerical experiments show the effectiveness and robustness of our approach, together with the speed-up obtained in mixed precision arithmetic, for a range of problem parameters and different Runge–Kutta methods.