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Center for Nonlinear Studies |
The physical model that is our main concern is the equations of fluid dynamics written in Lagrangian form. Being given a two-dimensional unstructured grid, we derive a staggered discretization of the hydrodynamics equations. The term staggered refers to spatial centering in which position, velocity and kinetic energy are centered at nodes, while thermodynamical variables are located at cell centers. The staggered discretization is performed following the sub-cell forces formalism introduced by Caramana, Burton, Shashkov and Whalen in JCP 146 (1998) 227--262. This formalism provides the rigorous compatibility between the discrete forces and their corresponding works so that total energy conservation is guaranteed to roundoff error. In traditional staggered discretization, the dissipation of kinetic energy into internal energy is ensured by the addition of an ad-hoc artificial viscosity term, whose formulation is far from being obvious.
In the present work, we propose an alternative approach to ensure entropy production through shock waves. To this end, we construct viscous tensorial sub-cell forces which are proportional to the jump between the node and the cell center velocity. The cell center velocity is naturally computed by using a two-dimensional approximate Godunov solver located at the cell centers. This Godunov solver is very similar to the one developed in our recent paper JCP 228 (2009) 2391--2425. This new framework should provide a deeper understanding of the relations between staggered and recent cell-centered Lagrangian schemes. Our staggered discretization, in its two-dimensional Cartesian version, not only conserves momentum and total energy but also satisfies a local entropy inequality.
Classical numerical tests are presented in order to assess the robustness and the accuracy of this new scheme.
Host: Mikhail Shashkov