Pseudotransient continuation and differential-algebraic equations

Pseudotransient continuation is a practical technique for globalizing the computation of steady-state solutions of nonlinear differential equations. The technique employs adaptive time-stepping to integrate an initial value problem derived from an underlying ODE or PDE boundary value problem until sufficient accuracy in the desired steady-state root is achieved to switch over to Newton's method and gain a rapid asymptotic convergence. The existing theory for pseudotransient continuation includes a global convergence result for differential equations written in semidiscretized method-of-lines form. However, many problems are better formulated or can only sensibly be formulated as differential-algebraic equations (DAEs). These include systems in which some of the equations represent algebraic constraints, perhaps arising from the spatial discretization of a PDE constraint. Multirate systems, in particular, are often formulated as differential-algebraic systems to suppress fast time scales (acoustics, gravity waves, Alfven waves, near equilibrium chemical oscillations, etc.) that are irrelevant on the dynamical time scales of interest. In this paper we present a global convergence result for pseudotransient continuation applied to DAEs of index 1, and we illustrate it with numerical experiments on model incompressible flow and reacting flow problems, in which a constraint is employed to step over acoustic waves.