Newton-Like Pseudo-Arclength Methods for Computing Simple Turning Points

TL;DR: A new method for computing simple turning points of nonlinear equations of the form G(u,\lambda ) = 0 which is based on applying Newton's method to the characterization of digma, where $\sigma $ is a pseudo-arclength parameter used in a continuation method for following the solution paths.

Abstract: We present a new method for computing simple turning points of nonlinear equations of the form $G(u,\lambda ) = 0$ which is based on applying Newton's method to the characterization ${{d\lambda (\sigma )} / {d\sigma }} = 0$, where $\sigma $ is a pseudo-arclength parameter used in a continuation method for following the solution paths. The method is quadratically convergent and needs only one starting point on the solution path. Second derivatives of G (or difference approximations of them) have to be computed but the method is relatively insensitive to their values and they also give rise to a more accurate second order predictor in the continuation method. We present a chord-Newton variant for improving the efficiency of the algorithm which requires only one factorization of a Jacobian matrix. We also present a damped-Newton variant for improving the robustness and the global convergence of the algorithm. Results of numerical experiments on two standard nonlinear elliptic problems of Simpson's [SIAM J. Numer. Anal., 12 (1975), pp. 439–451] show that the new algorithm compares favorably with the best of the existing methods in terms of efficiency and robustness.