A Parallel Adaptive Nonlinear Elimination Preconditioned Inexact Newton for Transonic Full Potential Flow Problems

Abstract

The inexact Newton method (INB) is the one of the most commonly-used methods for solving large spare nonlinear systems of equations arising from the discretization of partial differential equations. The method is quite robust and efficient for the smooth nonlinear systems, but for the case that the solution to the nonlinear system or its derivatives with local discontinuity, the convergence rate of INB degrades, at worst, it fails to converge even using some globalization technique. Such local discontinuity causes the nonlinear system unbalanced and the problem often appear in a variety of applications in computational fluid mechanics, such as the shock wave in the transonic compressible flows, the boundary layers or singularity around the corner in the high-Reynolds number incompressible flows.The recent research efforts for the INB-based nonlinear solver are to develop the effective nonlinear preconditioning to enhance its robustness and efficiency. As for linear systems, a nonlinear preconditioner can be applied to the nonlinear system on the left (referred as left preconditioning) or on the right (referred as right nonlinear preconditioning). The basic idea of left nonlinear preconditioning is to reformulate implicitly the original system as a new balanced system by some mean and then solve the new easier nonlinear system by INB. The additive Schwarz preconditioned inexact Newton algorithm (ASPIN) is belong to this class, which constructs the nonlinear preconditioner based on the Schwarz framework.On the other hand, the philosophy of right preconditioning is different, in which the nonlinear function does not need to be changed. For example, Hwang et. al successfully employ nonlinear elimination as right nonlinear preconditioner for INB with application in quasi one- dimensional shocked duct flow calculation. The basic idea of NE is to implicitly remove those components causing troubles and obtain a better balanced intermediate solution for which INB can be applied. In that paper, the assumption made is the components to be eliminated are known in advance based on the physical knowledge for the problem, which corresponds to the region near the nozzle. However, in practice, it might not be always possible to determine the bad components need to be eliminated, especially for the multi-dimensional applications.In this work, we propose a new algorithm, namely a parallel adaptive nonlinear elimination inexact Newton method (INB-adaptive NE) that relaxes such impractical assumption. We use the global information obtained by an intermediate INB solution to define adaptively the to-be- eliminated components. Taking the case of transonic flows passing the airfoil as a example, we report the numerical results obtained by a cluster of PCs that through the information exchange between the global and local data, the efficiency and robustness of INB can be greatly improved. Furthermore, INB-adaptive NE is nonlinearly scalable with respect to the number of processors similar to ASPIN. Due to the flexibility for the selection of the global Jacobian solver, the linear scalability of the method can be easily taken care by multilevel Schwarz Krylov subspace method.