A finite difference discretization method for heat and mass transfer with Robin boundary conditions on irregular domains

This paper proposes a finite difference discretization method for simulations of heat and mass transfer with Robin boundary conditions on irregular domains. The level set method is utilized to implicitly capture the irregular evolving interface, and the ghost fluid method to address variable discontinuities on the interface. Special care has been devoted to providing ghost values that are restricted by the Robin boundary conditions. Specifically, it is done in two steps: 1) calculate the normal derivative in cells adjacent to the interface by reconstructing a linear polynomial system; 2) successively extrapolate the normal derivative and the ghost value in the normal direction using a linear partial differential equation approach. This method produces second-order accurate solutions for both the Poisson and heat equations with Robin boundary conditions, and first-order accurate solutions for the Stefan problems. The solution gradients are of first-order accuracy, as expected. It is easy to implement in three-dimensional configurations, and can be straightforwardly generalized into higher-order variants. The method thus represents a promising tool for practical heat and mass transfer problems involving Robin boundary conditions.