Staggered Lagrangian Discretization Based on Cell-Centered Riemann Solver and Associated Hydrodynamics Scheme

Abstract

The aim of the present work is to develop a general formalism to derive staggered discretizations for Lagrangian hydrodynamics on two-dimensional unstructured grids. To this end, we make use of the compatible discretization that has been initially introduced by E. J. Caramana et al., in J. Comput. Phys., 146 (1998). Namely, momentum equation is discretized by means of subcell forces and specific internal energy equation is obtained using total energy conservation. The main contribution of this work lies in the fact that the subcell force is derived invoking Galilean invariance and thermodynamic consistency. That is, we deduce a general form of the sub-cell force so that a cell entropy inequality is satisfied. The subcell force writes as a pressure contribution plus a tensorial viscous contribution which is proportional to the difference between the nodal velocity and the cell-centered velocity. This cell-centered velocity is a supplementary degree of freedom that is solved by means of a cell-centered approximate Riemann solver. To satisfy the second law of thermodynamics, the local subcell tensor involved in the viscous part of the subcell force must be symmetric positive definite. This subcell tensor is the cornerstone of the scheme. One particular expression of 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. The results obtained for various representative configurations of one and two-dimensional compressible fluid flows show the robustness and the accuracy of this scheme.

A Conservative Lagrangian Scheme for Solving Compressible Fluid Flows with Multiple Internal Energy Equations

“…For the staggered-grid based Lagrangian method, the algorithms are built on a staggered discretization in which velocity (momentum) is stored at vertices, while density and internal energy are stored at cell centers. The density/internal energy and velocity are solved on two different control volumes directly, see, e.g., [1,3,4,6,25,29]. An artificial viscosity term [5,7,29] is usually added to the scheme to prevent spurious oscillations near the discontinuities.…”

A 3D finite element ALE method using an approximate Riemann solution

“…The method follows the finite volume works in [27][28][29][30] by solving a multidirectional Riemann-like problem at the cell center. The method follows the finite volume works in [27][28][29][30] by solving a multidirectional Riemann-like problem at the cell center.…”

“…The multidimensional Riemann-like problems were first proposed for Lagrangian CCH [10][11][12]26]; since then, researchers have extended these Riemann problems to finite volume SGH [27][28][29][30] and edge-based finite element PCH [16,21]. The multidimensional Riemann-like problems were first proposed for Lagrangian CCH [10][11][12]26]; since then, researchers have extended these Riemann problems to finite volume SGH [27][28][29][30] and edge-based finite element PCH [16,21].…”

A 3D staggered Lagrangian scheme for ideal magnetohydrodynamics on unstructured meshes

“…Here we use the classical discretization of momentum equation in the staggered scheme by inducing the subcell force F cp , which was introduced in the work of Maire et al, and then, the semidiscrete momentum equation over the dual cell Ω p is here, F cp is the subcell force which acts from subcell Ω cp onto node p . We can see the above momentum Equation as the second Newton law applied to a particle of mass m p moving with velocity V p .…”

“…Here, M cp is the subcell matrix which controls the velocity jump between the nodal and cell‐centered velocity. From the form of the subcell force F cp , it can turn in to the classic formalism for hydrodynamics in the work of Maire et al without the effect of a magnetic field, and this F cp ensures the entropy inequality in a cell at the semidiscrete level. Substituting into , where is a symmetric positive definite matrix.…”