Conditions for quadratic convergence of quick periodic steady-state methods

Abstract

The periodic steady-state solution of nonlinear dynamic systems can be computed by the so-called quick steady-state methods, e.g., Newton iteration, optimization or extrapolation. These methods compute a vector z(t_{0}) of the periodic steady-state solution by requiring the solution to repeat itself periodically, z(t_{0} + T) = z(t_{0}) where T is the period. It is essential for the performance of the quick steady-state methods that the solution of the differential equations at t_{0} + T is a smooth function of the initial value at t_{0} . This paper gives sufficient conditions for the numerical approximation of the solution at t_{0} + T to be differentiable with Lipschitz continuous derivative with respect to the initial vector at t_{0} . The numerical approximation is obtained by the backward differentiation formulas, and two cases are considered. First the nonlinear algebraic corrector equations are solved exactly, and this case deals both with continuously differentiable problems and piecewise-linear problems. Secondly the algebraic equations are solved to an accuracy consistent with the discretization error of the backward differentiation formulas. Finally, the theorems are illustrated by examples.