Pseudo-Transient Continuation for Nonsmooth Nonlinear Equations

Abstract

Pseudo-transient continuation is a Newton-like iterative method for computing steady-state solutions of differential equations in cases where the initial data are 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 PDE with a Lipschitz continuous, but nondifferentiable, 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 are not designed to find steady-state solutions of time-dependent problems. The method in this paper synthesizes the ideas from nonsmooth calculus and the method of pseudo-transient continuation.