Study on the second-order additional source term method for handling boundary conditions

An additional source term is an effective way to deal with boundary conditions to accelerate convergence rate, and it has been widely used in the literature. However, we find that this method has only first-order accuracy in its present form, which may decrease the accuracy of the numerical solution. A second-order additional source term method is proposed in this article, with detailed derivations. The numerical examples show that, compared to the first-order additional source term method, the second-order one has higher precision. Moreover, a unified expression is presented to treat the three common types of boundary conditions.