This paper discusses the use of two variations on Newton's method, quasi-Newton and full-Newton, for the solution of the Euler equations on unstructured triangular grids. The ILU(n)-preconditioned GMRES algorithm is employed in the solution of the Jacobian matrix problem which arises at each iteration. In the quasi-Newton method, a rst-order approximation to the Jacobian matrix is used with the standard GMRES implementation. Both standard and matrix-free implementations of GM-RES are used in the full-Newton scheme, and the latter is shown to be much faster and more e cient. A hybrid scheme is presented which makes use of the strengths of both full-and quasi-Newton implementations, resulting in very fast convergence to steady state. Finally, optimal preconditioning and reordering strategies are presented.
A fast Newton-Krylov algorithm is presented that solves the turbulent Navier-Stokes equations on unstructured 2-D grids. The model of Spalart and Allmaras provides the turbulent viscosity and is loosely coupled to the mean-flow equations. It is often assumed that the turbulence model must be fully coupled to obtain the full benefit of an inexact Newton algorithm. We demonstrate that a loosely coupled algorithm is effective and has some advantages, such as reduced storage requirements and smoother transient oscillations. A transonic single-element case converges to 1 10 12 in 90 s on recent commodity hardware, whereas the lift coefficient is converged to three figures in one quarter of that time.
A fast Newton-Krylov algorithm is presented that solves the turbulent Navier-Stokes equations on unstructured 2-D grids. The model of Spalart and Allmaras provides the turbulent viscosity and is loosely coupled to the mean-flow equations. It is often assumed that the turbulence model must be fully coupled to obtain the full benefit of an inexact Newton algorithm. We demonstrate that a loosely coupled algorithm is effective and has some advantages, such as reduced storage requirements and smoother transient oscillations. A transonic single-element case converges to 1 10 12 in 90 s on recent commodity hardware, whereas the lift coefficient is converged to three figures in one quarter of that time.
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