Multiple solutions for steady differential equations via hyperspherical path-tracking of homotopy curves

A multiple solutions finder method for steady Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs) is designed combining the classical finite differences discretization approach and homotopy continuation with hyperspherical pathtracking. The proposed methodology was independently validated employing a reported multiple solutions ODE, and a designed non-linear 2-D PDE with two solutions. In this work, hyperspherical path-tracking of homotopy curves is consistently employed as an effective strategy for computing unreported multiple solution vectors for an elliptic system of 2-D PDEs for natural convection. All the solutions found are mesh-size independent and mathematically satisfactory, thence they are proposed as benchmark for solver methods of numerical nonlinear algebraic systems applied on PDEs and ODEs with multiple steady states. (C) 2019 Elsevier Ltd. All rights reserved.