Abstract
In this paper we implement the element-by-element preconditioner and inexact Newton-Krylov methods (developed in the past) for solving stabilized computational fluid dynamics (CFD) problems with spectral methods. Two different approaches are implemented for speeding up the process of solving both steady and unsteady incompressible Navier-Stokes equations. The first approach concerns the application of a scalable preconditioner namely the element by element LU preconditioner, while the second concerns the application of Newton-Krylov (NK) methods for solving non-linear problems. We obtain good agreement with benchmark results on standard CFD problems for various Reynolds numbers. We solve the Kovasznay flow and flow past a cylinder at Re-$100$ with this approach. We also utilize the Newton-Krylov algorithm to solve (in parallel) important model problems such as flow past a circular obstacle in a Newtonian flow field, three dimensional driven cavity, flow past a three dimensional cylinder with different immersion lengths. We explore the scalability and robustness of the formulations for both approaches and obtain very good speedup. Effective implementations of these procedures demonstrate for relatively coarse macro-meshes<br />the power of higher order methods in obtaining highly accurate results in CFD. While the procedures adopted in the paper have been explored in the past the novelty lies with applications with higher order methods which have been known to be computationally intensive.
Highlights
In both computational research and practical applications there is continuing interest in solving incompressible fluid flow problems efficiently
This stabilization of the Galerkin formulation presented in this paper is a generalization of the Galerkin Least squares (GLS) formulation, and the SUPS procedure employed for incompressible flows
Inexact Newton-Krylov Methods Upon implementaton of the element by element (EBE)-LU preconditioner for solving the incompressible Navier-Stokes equations, we further explore using non-linear solvers for both two-and three-dimensional incompressible flows
Summary
In both computational research and practical applications there is continuing interest in solving incompressible fluid flow problems efficiently. The pressures are post computed from the velocities via the incompressibility constraint Using such techniques to solve complex problems are beset with difficulties primarily resulting from the high condition numbers associated with the discrete form of the penalty terms which can engender convergence issues with high values of the penalty parameter. Navier-Stokes equations, and a symmetric positive definite matrix, which can be solved with conjugate gradient solvers combining Jacobi or multi-grid preconditioning (Quarteroni, 1999) Another advantage of the LSFEM is that it provides a parameter free formulation for solving incompressible flow. To control the relatively high computation times for solving complicated problems, one has to resort to non-linear solvers and effective preconditioning techniques In this context using effective preconditioners and Newton-Krylov algorithms are of interest for solving large linear systems generated from spectral discretizations of the Navier-Stokes equations
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