Nonlinear preconditioning applied to internal flows

Abstract

Solution of large and sparse nonlinear system is required in many computational science and engineering areas. Well-known inexact Newton method converges fast with a good initial guess, however such an initial guess is hard to obtain for the problems with unbalanced nonlinearity, such as fluid flow problems with a high Reynolds number. Nonlinear preconditioning has been proposed to deal with this issue by constructing a preconditioned nonlinear system and solving it by inexact Newton. This approach is regarded as left nonlinear preconditioner since it changes the nonlinear system. On the other hand, right nonlinear preconditioner eliminates some previously chosen compo nents of the unknowns to form a better approximate solution to participate in inexact Newton iterations. We consider three preconditioning methods: additive Schwarz pre conditioned inexact Newton (ASPIN) which is based on domain decomposition meth ods, field-split preconditioned inexact Newton (FSPIN) which splits the components and solves them in an additive Schwarz manner, and nonlinear elimination precondi tioned inexact Newton (NEPIN). In this thesis, various studies have been conducted to observe the behavior of the algorithms with respect to two different implementa tions of finite difference method on boundaries, some tolerance parameters and some algorithmic parameters for steady-state lid-driven cavity problem at several values of Reynolds number. Our numerical tests obtained on parallel show that NEPIN is more robust than inexact Newton and other nonlinear preconditioning methods.