Globally convergent algorithms for nonsmooth nonlinear equations in computational fluid dynamics

Coffey, Todd; McMullan, Richard J.; Kelley, C. T.; Mcrae, D. S.

doi:10.1016/s0377-0427(02)00697-0

Cited by 7 publications

(17 citation statements)

References 22 publications

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“…In this paper we extend the theoretical convergence results of [7,18] to problems with certain nonsmooth nonlinearities and, thereby, partially explain the results reported in [8,10]. We also show how generalized derivatives can be approximated by finite differences, and how those approximate derivatives can be used effectively both in locally convergent iterations, such as those which arise in temporal integration, and in the context of Ψtc .…”

Section: Introductionsupporting

confidence: 72%

“…Ψtc is a predictor-corrector method for efficient integration of a time-dependent differential equation to steady state. The objective of the method is not temporal accuracy, but rather to resolve the transient behavior of the solution until the iteration is close to steady state, and then to increase the "time step" and transition to a fast Newton-like method.In this paper we extend the theoretical convergence results of [7,18] to problems with certain nonsmooth nonlinearities and, thereby, partially explain the results reported in [8,10]. We also show how generalized derivatives can be approximated by finite differences, and how those approximate derivatives can be used effectively both in locally convergent iterations, such as those which arise in temporal integration, and in the context of Ψtc .…”

supporting

confidence: 72%

“…tions, such as line searches, can fail as commonly implemented in practice for both smooth and nonsmooth equations [7,8,18]. The existing global convergence results for nonsmooth nonlinear equations are either based on line searches for a inexact Newton formulation [9,22], a sequence of smooth approximations [4,29], or explicit treatment of the nonsmoothness [20].…”

Section: Performing Organization Name(s) and Address(es)mentioning

confidence: 99%

“…Possible failure modes [8] are stagnation of the iteration at a singularity of F , the Jacobian of F , or or finding a solution other than x * .…”

Section: Pseudo-transient Continuationmentioning

confidence: 99%

“…In [7,8,10,18] we report on compuational results that show that both the local and global phases of the Ψtc iteration perform as (2.11) predicts even if the nonlinearity is nonsmooth [7,8,10] and the derivative is approximated by differencing [10,13,14,30].…”

Section: Pseudo-transient Continuationmentioning

confidence: 99%

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Pseudo-Transient Continuation for Nonsmooth Nonlinear Equations

Fowler¹,

Kelley²

2005

SIAM J. Numer. Anal.

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Abstract. Pseudo-transient continuation is a Newton-like iterative method for computing steady-state solutions of differential equations in cases where the initial data is far from a steady state. The iteration mimics a temporal integration scheme, with the time step being increased as steady state is approached. The iteration is an inexact Newton iteration in the terminal phase.In this paper we show how steady-state solutions to certain ordinary and differential algebraic equations with nonsmooth dynamics can be computed with the method of pseudo-transient continuation. An example of such a case is a discretized partial differential equation with a Lipschitz continuous, but non-differentiable, constitutive relation as part of the nonlinearity. In this case we can approximate a generalized derivative with a difference quotient.The existing theory for pseudo-transient continuation requires Lipschitz continuity of the Jacobian. Newton-like methods for nonsmooth equations have been globalized by trust-region methods, smooth approximations, and splitting methods in the past, but these approaches require problem-specific components in an algorithm. The method in this paper addresses the nonsmoothness directly.Key words. Pseudo-transient Continuation, Nonlinear equations, Semismooth functions, Clarke differential AMS subject classifications. 65H10, 65H20, 65L05, 1. Introduction. In this paper we show how pseudo-transient continuation (Ψtc ) can be used to solve a class of nonsmooth nonlinear equations. Ψtc is a predictor-corrector method for efficient integration of a time-dependent differential equation to steady state. The objective of the method is not temporal accuracy, but rather to resolve the transient behavior of the solution until the iteration is close to steady state, and then to increase the "time step" and transition to a fast Newton-like method.In this paper we extend the theoretical convergence results of [7,18] to problems with certain nonsmooth nonlinearities and, thereby, partially explain the results reported in [8,10]. We also show how generalized derivatives can be approximated by finite differences, and how those approximate derivatives can be used effectively both in locally convergent iterations, such as those which arise in temporal integration, and in the context of Ψtc . This aspect of the work is motivated by several papers on simulation of unsaturated flow, [10,14,15,24,30,31], in which Lipschitz continuous spline approximations to the non-Lipschitz continuous van Geneuchten and Mualem [25,33] constitutive laws are used. These nonsmooth functions are then differentiated with finite differences as if they were smooth. The results in this paper explain the success reported in those papers. Another aspect of the paper is an extension of the local results in [9,21,27,28].Ψtc methods are particularly appropriate for the types of nonsmooth nonlinearities which we discuss in this paper. Traditional methods for globalizing iterative methods for nonlinear equa-

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Section: Introductionsupporting

confidence: 72%

supporting

confidence: 72%

Section: Performing Organization Name(s) and Address(es)mentioning

confidence: 99%

“…Possible failure modes [8] are stagnation of the iteration at a singularity of F , the Jacobian of F , or or finding a solution other than x * .…”

Section: Pseudo-transient Continuationmentioning

confidence: 99%

Section: Pseudo-transient Continuationmentioning

confidence: 99%

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Pseudo-Transient Continuation for Nonsmooth Nonlinear Equations

Fowler¹,

Kelley²

2005

SIAM J. Numer. Anal.

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Pseudo‐transient continuation for nonlinear transient elasticity

Gee

Kelley

Lehoucq

2008

Numerical Meth Engineering

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SUMMARYThis paper demonstrates how pseudo-transient continuation improves the efficiency and robustness of a Newton iteration within a non-linear transient elasticity simulation. Pseudo-transient continuation improves efficiency by enabling larger time steps than possible with a Newton iteration. Robustness improves because pseudo-transient continuation recovers the convergence of Newton's method when the initial iterate is not within the region of local convergence. We illustrate the benefits of pseudo-transient continuation on a non-linear transient simulation of a buckling cylinder, including a comparison with a line search-based Newton iteration.

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Pseudotransient Continuation for Solving Systems of Nonsmooth Equations with Inequality Constraints

Chen

2010

J Optim Theory Appl

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Globally convergent algorithms for nonsmooth nonlinear equations in computational fluid dynamics

Cited by 7 publications

References 22 publications

Pseudo-Transient Continuation for Nonsmooth Nonlinear Equations

Pseudo-Transient Continuation for Nonsmooth Nonlinear Equations

Pseudo‐transient continuation for nonlinear transient elasticity

Pseudotransient Continuation for Solving Systems of Nonsmooth Equations with Inequality Constraints

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