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SIAM J. SCI. COMPUT. c 2007 Society for Industrial and Applied Mathematics
Vol. 29, No. 4, pp. 17811824
A CELL-CENTERED LAGRANGIAN SCHEME FOR
TWO-DIMENSIONAL COMPRESSIBLE FLOW PROBLEMS
PIERRE-HENRI MAIRE, R´EMI ABGRALL, J´ER^OME BREIL, AND JEAN OVADIA§
Abstract. We present a new Lagrangian cell-centered scheme for two-dimensional compressible
flows. The primary variables in this new scheme are cell-centered, i.e., density, momentum, and
total energy are defined by their mean values in the cells. The vertex velocities and the numerical
fluxes through the cell interfaces are not computed independently, contrary to standard approaches,
but are evaluated in a consistent manner due to an original solver located at the nodes. The main
new feature of the algorithm is the introduction of four pressures on each edge, two for each node
on each side of the edge. This extra degree of freedom allows us to construct a nodal solver which
fulfills two properties. First, the conservation of momentum and total energy is ensured. Second, a
semidiscrete entropy inequality is provided. In the case of a one-dimensional flow, the solver reduces
to the classical Godunov acoustic solver: it can be considered as its two-dimensional generalization.
Many numerical tests are presented. They are representative test cases for compressible flows and
demonstrate the robustness and the accuracy of this new solver.
Key words. Godunov-type schemes, hyperbolic conservation laws, Lagrangian gas dynamics
AMS subject classifications. 65M06, 76L05, 76N05