Pierre-Henri Maire

CEA/DAM CESTA – 🇫🇷 Le Barp (France)

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On Nodal Flux Approximations for Multidimensional Cell-Centered Lagrangian Hydrodynamics

We revisit the cell-centered Lagrangian scheme EUCCLHYD, introduced in [Maire et al., SISC 2007], and reinterpret it as a multipoint Finite Volume method. Each cell face is split into subfaces attached to the nodes. On each subface, the flux comes from a simple approximate Riemann solver, parametrized by the normal component of the intermediate velocity. This solver is positivity preserving but not conservative in the classical sense: the left- and right-sided fluxes differ. Conservation is restored by a node-based balance, which acts as a multidimensional consistency condition. Closing the free parameter with a nodal velocity turns this balance into the nodal solver, which is automatically compatible with the Geometric Conservation Law. The resulting scheme is positivity preserving under an explicit time step condition, and satisfies a semi-discrete entropy inequality. We also show that the node-based balance is necessary and sufficient for a unique flux to exist on each half-face. This flux depends on all the states surrounding the node, hence the name multipoint flux. Numerical results on the Sod, Sedov and Noh problems illustrate the approach. The same framework underlies recent Eulerian extensions in 2D and 3D.