Robust continuation methods for tracing solution curves of parameterized systems

The continuation methods are efficient methods to trace solution curves of nonlinear systems with parameters, which are common in many fields of science and engineering. Existing continuation methods are unstable for some complicated cases in practice, such as the case that solution curves are close to each other or the case that the curve turns acutely at some points. In this paper, a more robust corrector strategy—sphere corrector is presented. Using this new strategy, combining various predictor strategies and various iterative methods with local quadratic or superlinear convergence rates, robust continuation procedures for tracing curves are given. When the predictor steplength is no more than the so-called granularity of solution curves, our procedure of tracing solution curve can avoid “curve-jumping” and trace the whole solution curve successfully. Numerical experiments illustrate our method is more robust and efficient than the existing continuation methods.