A study of the approximation of an interface with the height function method

In this study we introduce the continuous height function to study the properties of its discrete version as an approximation of an interface line. We show that the continuous height function is a second-order accurate approximation of the interface. The slope and the curvature of the interface line computed with a centered finite differences scheme based on the height function are shown to be second-order accurate as well. The reconstruction error is proved to be second-order with a linear interface reconstruction and third-order with a parabolic approximation of the interface. We then discuss a few implementation issues about the stencil to be used to compute the height function and the data interpolation in the grid. We conclude the paper with some numerical tests for a circular interface.