An advection velocity correction scheme for interface tracking using the level-set method

We present an advection velocity correction (AVC) scheme for interface tracking using the level-set method in this paper. The key idea is to apply a correction to the interface advection velocity at points adjacent to the zero level-set, so as to enforce the preservation of the signed distance function property at these points. As such, the AVC scheme eliminates the need for explicit sub-cell fix approaches as reinitialization at points adjacent to the zero level-set is not needed. This approach of correcting the advection velocity field near the interface and computing the signed distance function; SDF to a high order of accuracy near the interface, rather than applying an explicit sub-cell fix during the reinitialization step represents the key novel aspect of the AVC scheme. We present results from using the AVC scheme along with advection and reinitialization schemes using upwind finite differencing on uniform meshes in this paper. These results are determined for four canonical test problems: slotted disk rotation, deforming sphere, interacting circles and vortex in a box. We compare these results with corresponding results determined using a recently proposed explicit sub-cell fix based reinitialization scheme (CR2). These comparisons show that the AVC scheme yields significantly improved conservation of enclosed volume/area within the interface. Note, the present AVC scheme achieves this by only modifying velocity field values at mesh points. Therefore, the AVC algorithm can in principle be used within the framework of nearly any numerical scheme used to compute interface evolution using the level-set method, even on non-uniform and unstructured meshes, in order to achieve improvements in solution quality.