Abstract
Numerical continuation is the process of solving nonlinear equations of the form G(u, L)=0 for various parameter values, L. We discuss established numerical continuation techniques for solution paths of G(u, L)=0 containing regular points and simple folds. Pseudo-arclength continuation is a widely used technique that no longer solves G(u, L)=0 directly, but solves a newly parameterized version F(u(s), L(s))=0 where s is arclength. We present a new characterization for certain classes of points that leads to an upper bound on the norm of the inverse of F’. This bound is needed to meaningfully quantify the convergence of Newton's method in the context of pseudo-arclength continuation.
In particle fluids, it is important to accurately predict conditions under which the structure and thermodynamics of a fluid change. Of particular interest are phase transitions, for instance, when water turns from liquid to vapor as a function of temperature. Such information can often be obtained through rigorous and accurate continuation studies, yet is historically burdensome to produce computationally. To this end, we present new integral equation theory that model atomic and molecular fluids developed by the Institute of Molecular Design at the University of Houston. We show cases for which this new theory results in more accurate fluid structure and thermodynamics than previously reported in the literature. We discuss the numerical continuation problems of interest for both atomic and molecular fluids and how to implement them using this new theory and software developed at Sandia National Laboratories. Employing these theoretical and computational tools, we provide evidence for the potential to uncover new chemical and physical properties of fluids in an accurate and efficient way.
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