PID adaptive control of incremental and arclength continuation in nonlinear applications

SUMMARY A proportional-integral-derivative (PID) control approach is developed, implemented and investigated numerically in conjunction with continuation techniques for nonlinear problems. The associated algorithm uses PID control to adapt parameter stepsize for branch—following strategies such as those applicable to turning point and bifurcation problems. As representative continuation strategies, incremental Newton, Euler‐Newton and pseudo-arclength continuation techniques are considered. Supporting numerical experiments are conducted for finite element simulation of the ‘driven cavity’ Navier‐Stokes benchmark over a range in Reynolds number, the classical Bratu turning point problem over a reaction parameter range, and for coupled fluid flow and heat transfer over a range in Rayleigh number. Computational performance using PID stepsize control in conjunction with inexact Newton‐Krylov solution for coupled flow and heat transfer is also examined for a 3D test case. Copyright q 2009 John Wiley & Sons, Ltd.