Toward a higher order unsteady finite volume solver based on reproducing kernel methods

During the last decades, research efforts are headed to develop high order methods on CFD and CAA to reach most industrial applications (complex geometries) which need, in most cases, unstructured grids. Today, higher-order methods dealing with unstructured grids remain in infancy state and they are still far from the maturity of structured grids-based methods when solving unsteady cases. From this point of view, the development of higher order methods for unstructured grids become indispensable. The finite volume method seems to be a good candidate, but unfortunately it is difficult to achieve space flux derivation schemes with very high order of accuracy for unsteady cases. In this paper we propose, a high order finite volume method based on Moving Least Squares approximations for unstructured grids that is able to reach an arbitrary order of accuracy on unsteady cases. In order to ensure high orders of accuracy, two strategies were explored independently: (1) a zero-mean variables reconstruction to enforce the mean order at the time derivative and (2) a pseudo mass matrix formulation to preserve the residuals order.