A parallel nonlinear additive Schwarz preconditioned inexact Newton algorithm for incompressible Navier–Stokes equations

Hwang, Feng Nan; Cai, Xiao‐Chuan

doi:10.1016/j.jcp.2004.10.025

Cited by 69 publications

(41 citation statements)

References 30 publications

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“…This technique is widely-used in the context of partial differential equations as reported in many articles in the proceedings on the annually conferences on domain decomposition methods. A cheap and fast variant, the Restricted Additive Schwarz Method (RAS, RASM) by Cai and Sarkis [13], that is also integrated in the PETSc library [2], has been successfully combined with a NewtonKrylov method within the context of flow solvers [12,19,22,23]. As demonstrated in [19] the RAS method can also be combined with the PBILU(0) algorithm for the subdomains.…”

Section: Discussionmentioning

confidence: 99%

“…While the RAS method is used as preconditioner for the linear systems in [13], Schwarz preconditioners can also be used as a preconditioner for the non-linear systems of equations as presented in [11]. This non-linear technique was applied to a one-dimensional compressible flow, denoted by "additive Schwarz preconditioned inexact Newton method" (ASPIN), in [12] and has also been successfully used in [23]. ASPIN is a non-linear block-Jacobi iteration followed by a Newton linearization.…”

Section: Discussionmentioning

confidence: 99%

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Iterative Solvers for Discretized Stationary Euler Equations

Pollul

Reusken

2010

Notes on Numerical Fluid Mechanics and Multidisciplinary Design

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In this paper we treat subjects which are relevant in the context of iterative methods in implicit time integration for compressible flow simulations. We present a novel renumbering technique, some strategies for choosing the time step in the implicit time integration, and a novel implementation of a matrix-free evaluation for matrix-vector products. For the linearized compressible Euler equations, we present various comparative studies within the QUADFLOW package concerning preconditioning techniques, ordering methods, time stepping strategies, and different implementations of the matrix-vector product. The main goal is to improve efficiency and robustness of the iterative method used in the flow solver.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Iterative Solvers for Discretized Stationary Euler Equations

Pollul

Reusken

2010

Notes on Numerical Fluid Mechanics and Multidisciplinary Design

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“…For problems with high nonlinearities, preconditioning techniques on the nonlinear level, such as the additive Schwarz preconditioned inexact Newton (ASPIN) methods [9,25,39], the nonlinear restricted Schwarz preconditioners [10,16], the nonlinear dual-domain decomposition methods [48], the nonlinear balancing domain decomposition by constraints methods [30], the nonlinear elimination methods [26,27,29], and the composite nonlinear algebraic methods [6], have received increasing attention in recent years. In particular, some efforts have been made in applying the ASPIN method to solve the two-phase flow problems [51,53].…”

Section: B595mentioning

confidence: 99%

“…However, when the nonlinearity of the system is severely imbalanced, the size of λ k could be very small and the convergence becomes much slower. Observed from many numerical experiments, the slow convergence or sometimes divergence is often determined by the variables of equations in the system with the highest nonlinearities [9,10,25]. In our proposed algorithm, a nonlinear elimination step is applied as a subproblem solver inside the global RS iteration to smooth out the "high nonlinearity."…”

Section: B595mentioning

confidence: 99%

Active-Set Reduced-Space Methods with Nonlinear Elimination for Two-Phase Flow Problems in Porous Media

Yang¹,

Yang²,

Sun³

2016

SIAM J. Sci. Comput.

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Abstract. Fully implicit methods are drawing more attention in scientific and engineering applications due to the allowance of large time steps in extreme-scale simulations. When using a fully implicit method to solve two-phase flow problems in porous media, one major challenge is the solution of the resultant nonlinear system at each time step. To solve such nonlinear systems, traditional nonlinear iterative methods, such as the class of the Newton methods, often fail to achieve the desired convergent rate due to the high nonlinearity of the system and/or the violation of the boundedness requirement of the saturation. In the paper, we reformulate the two-phase model as a variational inequality that naturally ensures the physical feasibility of the saturation variable. The variational inequality is then solved by an active-set reduced-space method with a nonlinear elimination preconditioner to remove the high nonlinear components that often causes the failure of the nonlinear iteration for convergence. To validate the effectiveness of the proposed method, we compare it with the classical implicit pressure-explicit saturation method for two-phase flow problems with strong heterogeneity. The numerical results show that our nonlinear solver overcomes the often severe limits on the time step associated with existing methods, results in superior convergence performance, and achieves reduction in the total computing time by more than one order of magnitude.

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“…The basic idea of nonlinearly preconditioned inexact Newton algorithms ( [4,9]) is to find the solution u ∈ R n of (1) by solving an equivalent system…”

Section: Nonlinear Additive Schwarz Preconditioned Inexact Newton Algmentioning

confidence: 99%

Nonlinear Overlapping Domain Decomposition Methods

Cai

2009

Lecture Notes in Computational Science and Engineering

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Summary. We discuss some overlapping domain decomposition algorithms for solving sparse nonlinear system of equations arising from the discretization of partial differential equations. All algorithms are derived using the three basic algorithms: Newton for local or global nonlinear systems, Krylov for the linear Jacobian system inside Newton, and Schwarz for linear and/or nonlinear preconditioning. The two key issues with nonlinear solvers are robustness and parallel scalability. Both issues can be addressed if a good combination of Newton, Krylov and Schwarz is selected, and the right selection is often dependent on the particular type of nonlinearity and the computing platform.

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A parallel nonlinear additive Schwarz preconditioned inexact Newton algorithm for incompressible Navier–Stokes equations

Cited by 69 publications

References 30 publications

Iterative Solvers for Discretized Stationary Euler Equations

Iterative Solvers for Discretized Stationary Euler Equations

Active-Set Reduced-Space Methods with Nonlinear Elimination for Two-Phase Flow Problems in Porous Media

Nonlinear Overlapping Domain Decomposition Methods

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