The calculation of second-order accurate curvatures using surface normal vectors

A simple method is proposed to evaluate the curvature of interfaces. The essence of the method is to treat the surface normal (SN) vectors as independent variables, and to integrate them separately. The curvature is then evaluated from the SN vectors. It is found that the curvature can be evaluated with uniform second-order accuracy for circles as small as two grid cells in diameter, which has never been reported before. In addition, the method is based on the numerical solution of the partial differential equations, so it can be straightforwardly extended to unstructured grid systems. Three possible methods to evaluate the curvature from the SN vectors have been compared. It turns out that only the curvature interpolated from that of the centroid to the center of an interface segment achieves the second-order accuracy.