Time-dependent nonlinear collocation method and stability analysis for natural convection problems.

A time-dependent nonlinear framework based on meshfree collocation is proposed for solving natural convection problems involving multi-phases, in which the third-order Runge-Kutta method is introduced for temporal discretization while the two-step Newton-Raphson method is adopted for nonlinear iteration. To reduce the number of field variables, the common stream function-velocity equation is not directly used; instead, Darcy's law is introduced so that a three-phase coupling system describing natural convection can be established in terms of the stream function, vorticity, and temperature. As the resulting system is highly nonlinear, especially with vorticity involved, obtaining satisfactory solutions remains a challenging task. In view of flexibility and local nature of the reproducing kernel shape functions, they are adopted as the foundation of the proposed framework. Additionally, the corresponding stability analysis is carried out. The efficacy of the framework is demonstrated by the benchmark examples. To further explore the avenue to solve more complicated nonlinear problems, a four-phase coupling system involving concentration in addition to the aforementioned three variables is investigated. It is shown that the proposed solution strategy is capable of untangling the multi-phase coupling problems with and without double-diffusion. [ABSTRACT FROM AUTHOR]