Hyperspherical path tracking methodology as correction step in homotopic continuation methods

摘要

Homotopic trajectories are constructed through the calculation of the discrete points that constitute a curve that is known as the homotopic path. with is mathematically the same algenbraic system that is mapped by the homotopy function at every point but exhibits an arbitrary variation in the homotopic parameter. Thus, to calculate the homotopic parameter, complex strategies are commonly used to define a steo size that favors convergence. However, we demonstrate that thses strateties cannot guarantee the numberical stability of the path tracking process and are also quite complicated to understand and implement if numberical procedures. In this work, an N+1 dimensional version of the canonical equation of the sphere was solved in conjunction with the homeotpic system. Thus, throuth the use of N+1 variables and N+1 equations, the problem is defined and geometrically closed. This method was named "hyperspherical path tracking". In a combined methodology and results section, we present some heuristic observations in the construction of novel convergence criterion for homotopic methods. In addition, numerical evidence of the stability and good behavor of the tracing hyperspheres is presented. In all the sovedl example systems, the solution vecotrs that have been prrviously reported by other aughors were localized using our mehtod. In some cases, additional solution vercotrs were found. In addition, our method was able to circumvent the numerical challenge that is presented in the construction of homptopic paths that are deformed by bounded homotopies. The solution of a system derived form one benchamark function of two variables with multiple minima is presented, and some conclusions were obtained for this application. Finally, our method found 15 solution vectors for a large and highly nonlinear algebraic system of equations, which was obtained through the discretization of the set of elliptic partial differential equations (PDEs) that govern the natural convection in a defferentially heated square cavity; these solutions head not been proviously reported.