Improved numerical schemes for monotonic conservative scalar advection : tackling mesh imprinting and numerical wetting

The French atomic energy commission (CEA, France) deploys specific numerical methods to carry out its hydrodynamics simulations under especially severe constraints : advection and deformation over long distances, isentropic evolutions and strong shocks, complex equations of state and material mixtures, coupling with other various and complex physics, large meshes requiring massively parallel computers, etc.In this context the numerical scheme GEEC (Geometry, Energy, and Entropy Compatible) was recently developed at CEA. GEEC is generally second-order in space and time, except for the advection operator relative to the mesh which is only first-order.The present work adds two techniques compatible with each other to improve the formulation of the advection operator with the main objective of reducing "mesh-imprinting" and "numerical wetting":(i) the "co-mesh" strategy corrects the anisotropic behavior of advection imprinted on the numerical solution and caused by the orientation of the grid with respect to the velocity field, which is also known as "grid orientation effect";(ii) the n-dimensional Slope-And-Bound (ND-SAB) reconstruction algorithm defines a straight forward second-order reconstruction technique, and can be seen as an alternative strategy to classical directional splitting methods. The ND-SAB technique is inspired by the 1D SAB limitation strategy, which introduces a "clipping" method for slope reconstructions with the goal of suppressing numerical wetting.Both techniques have been validated individually and combined on uniform Cartesian grids with simulations for constant velocities. Future works will include testing this novel form of a second-order advection operator fully compatible with GEEC in its multi-fluid direct-ALE formalism.