Imposing mixed Dirichlet-Neumann-Robin boundary conditions on irregular domains in a level set/ghost fluid based finite difference framework

Abstract

In this paper, an efficient, unified finite difference method for imposing mixed Dirichlet, Neumann and Robin boundary conditions on irregular domains is proposed, leveraging on our previous work [Chai et al., J. Comput. Phys. 400 (2020): 108890]. The level set method is applied to describe the arbitrarily-shaped interface, and the ghost fluid method is utilized to address the complex discontinuities on the interface. The core of this method lies in providing required ghost values under the restriction of mixed boundary conditions, which is done in a fractional-step way. Specifically, the normal derivative is calculated in the concerned subdomain by aid of a linear polynomial reconstruction, then the normal derivative and the ghost value are successively extrapolated to the other subdomain using a linear partial differential equation approach. A series of Poisson problems with mixed boundary conditions and a heat transfer test are performed to validate the method, highlighting its convergence accuracy in the L1 and L∞ norms. The method produces second-order accurate solutions with first-order accurate gradients, and is easy to implement in multi-dimensional configurations. In summary, the method represents a promising tool for imposing mixed boundary conditions, which will be applied to practical problems in future work.