Abstract
We present the second-order multidimensional staggered grid hydrodynamics residual distribution (SGH RD) scheme for Lagrangian hydrodynamics. The SGH RD scheme is based on the staggered finite element discretizations as in [V. A. Dobrev, T. V.Kolev, and R. N. Rieben, SIAM J. Sci. Comput. 34 (2012), pp. B606--B641]. However, the advantage of the residual formulation over classical FEM approaches consists in the natural mass matrix diagonalization which allows one to avoid the solution of the linear system with the global sparse mass matrix while retaining the desired order of accuracy. This is achieved by using Bernstein polynomials as finite element shape functions and coupling the space discretization with the deferred correction type timestepping method. Moreover, it can be shown that for the Lagrangian formulation written in nonconservative form, our RD scheme ensures the exact conservation of the mass, momentum, and total energy. In this paper, we also discuss construction of numerical viscosity approximations for the SGH RD scheme, allowing us to reduce the dissipation of the numerical solution. Thanks to the generic formulation of the staggered grid RD scheme, it can be directly applied to both single- and multimaterial and multiphase models. Finally, we demonstrate computational results obtained with the proposed RD scheme for several challenging test problems.
Abgrall, Rémi