A finite-volume discretisation of the eikonal equation on unstructured vertex-based data structures

The eikonal equation is widely used to compute minimum travelling times or distance fields in computational physics. Recently, solutions to the eikonal equation have been used to define the initial field to be evolved with the level-set method in interface propagation problems. A practical implementation working on a generic tetrahedral data-structure requires bespoke numerical treatments. This paper presents a systematic numerical investigation on the stability, convergence and accuracy of vertex-centred unstructured finite-volume discretisations of the eikonal equation. Careful handling of the upwinding is crucial to obtain stable solutions on distorted grids. A fully-implicit Newton algorithm proves beneficial for robustness but requires a Harten-like regularisation of the advective flux Jacobian, to be calibrated based on the maximum mesh edge length. A 2nd order MUSCL reconstruction is adopted to reduce the numerical dissipation. The accuracy of the approach is verified on a set of benchmark and realistic test cases, including a swept NACA0012 wing. The algorithm is compared to a Fast Marching Method library available in the literature ( scikit-fmm ), showing that a similar or higher level of accuracy is achieved on the considered test cases.