Staggered Lagrangian Discretization Based on Cell-Centered Riemann Solver and Associated Hydrodynamics Scheme (Communications in Computational Physics, 2011)
The aim of the present work is to develop a general formalism to derivestaggered discretizations for Lagrangian hydrodynamics on two-dimensional unstructuredgrids. To this end, we make use of the compatible discretization that has been initiallyintroduced by E. J. Caramana et al., in J. Comput. Phys., 146 (1998). Namely, momentumequation is discretized by means of subcell forces and specific internal energyequation is obtained using total energy conservation. The main contribution of thiswork lies in the fact that the subcell force is derived invoking Galilean invariance andthermodynamic consistency. That is, we deduce a general form of the sub-cell force sothat a cell entropy inequality is satisfied. The subcell force writes as a pressure contributionplus a tensorial viscous contribution which is proportional to the differencebetween the nodal velocity and the cell-centered velocity. This cell-centered velocity isa supplementary degree of freedom that is solved by means of a cell-centered approximateRiemann solver. To satisfy the second law of thermodynamics, the local subcelltensor involved in the viscous part of the subcell force must be symmetric positivedefinite. This subcell tensor is the cornerstone of the scheme. One particular expressionof this tensor is given. A high-order extension of this discretization is provided.Numerical tests are presented in order to assess the efficiency of this approach. Theresults obtained for various representative configurations of one- and two-dimensionalcompressible fluid flows show the robustness and the accuracy of this scheme.