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Volume 10, Issue 4
Staggered Lagrangian Discretization Based on Cell-Centered Riemann Solver and Associated Hydrodynamics Scheme

Pierre-Henri Maire, Raphaël Loubère & Pavel Váchal

Commun. Comput. Phys., 10 (2011), pp. 940-978.

Published online: 2011-10

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  • 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.

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@Article{CiCP-10-940, author = {}, title = {Staggered Lagrangian Discretization Based on Cell-Centered Riemann Solver and Associated Hydrodynamics Scheme}, journal = {Communications in Computational Physics}, year = {2011}, volume = {10}, number = {4}, pages = {940--978}, 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.170310.251110a}, url = {http://global-sci.org/intro/article_detail/cicp/7470.html} }
TY - JOUR T1 - Staggered Lagrangian Discretization Based on Cell-Centered Riemann Solver and Associated Hydrodynamics Scheme JO - Communications in Computational Physics VL - 4 SP - 940 EP - 978 PY - 2011 DA - 2011/10 SN - 10 DO - http://doi.org/10.4208/cicp.170310.251110a UR - https://global-sci.org/intro/article_detail/cicp/7470.html KW - AB -

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.

Pierre-Henri Maire, Raphaël Loubère & Pavel Váchal. (2020). Staggered Lagrangian Discretization Based on Cell-Centered Riemann Solver and Associated Hydrodynamics Scheme. Communications in Computational Physics. 10 (4). 940-978. doi:10.4208/cicp.170310.251110a
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