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 path-tracking. 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.

结合经典有限差分离散化方法和同伦连续性以及超球面路径跟踪，设计了一种适用于稳态常微分方程（ODE）和偏微分方程（PDE）的多解查找器方法。。使用报告的多个解决方案ODE和带有两个解决方案的设计的非线性2-P PDE，对所提出的方法进行了独立验证。在这项工作中，同构曲线的超球面路径跟踪一直被用作一种有效策略，用于计算自然对流的二维PDE椭圆系统的未报告多个解矢量。所发现的所有解决方案都是独立于网格大小的，并且在数学上令人满意，因此提出了它们作为应用在具有多个稳态的PDE和ODE上的数值非线性代数系统的求解器方法的基准。