Multidimensional staggered grid residual distribution scheme for Lagrangian hydrodynamics
This work provides a more efficient and accurate numerical method for Lagrangian hydrodynamics simulations, which is important for applications in fluid dynamics and multiphase flow modeling.
The authors present a second-order multidimensional staggered grid residual distribution scheme for Lagrangian hydrodynamics that avoids solving linear systems with a global sparse mass matrix while ensuring exact conservation of mass, momentum, and total energy. Numerical results on challenging test problems demonstrate the scheme's effectiveness.
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 [Dobrev et al., SISC, 2012]. 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 non-conservative form, our residual distribution 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 to reduce the dissipation of the numerical solution. Thanks to the generic formulation of the staggered grid residual distribution scheme, it can be directly applied to both single- and multimaterial and multiphase models. Finally, we demonstrate computational results obtained with the proposed residual distribution scheme for several challenging test problems.