A node conservative cell-centered finite volume method for solving multidimensional Euler equations over general unstructured grids

We are interested in the numerical simulation of hypersonic flows around vehicles characterized by complex geometry. As a first step to move in this direction, we present a robust and accurate cell-centered Finite Volume (FV) method for solving the three-dimensional compressible Euler equations over general unstructured grids. This FV approach relies on a novel positivity-preserving discretization of the multidimensional Euler equations, which leverages a partitioning of cell faces into subfaces impinging at the nodes. The subface flux approximation is derived from an approximate Riemann solver, which incorporates not only the mean values of the cells adjacent to the subface but also the velocity of the node from which the subface originates. The projection of the nodal velocity onto the unit normal vector of the subface can be interpreted as a parameter in this Riemann solver. Consequently, the resulting subface flux is not unique, leading to a lack of conservation in the classical sense. Conservation is restored by ensuring that the subface fluxes around a node sum to zero, which determines the nodal velocity. This innovative multipoint flux approximation approach seems to eliminate the numerical pathologies commonly encountered in classical face-based FV formulations. The space and time second-order extension of this FV approach is classically deduced by means of a monotonic piecewise linear reconstruction. The robustness and accuracy of this novel numerical method are assessed against various demanding representative test cases in 2D and 3D.