A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes

We present a high-order cell-centered Lagrangian scheme for solving the two-dimensional gas dynamics equations on unstructured meshes. A node-based discretization of the numerical fluxes for the physical conservation laws allows to derive a scheme that is compatible with the geometric conservation law (GCL). Fluxes are computed using a nodal solver which can be viewed as a two-dimensional extension of an approximate Riemann solver. The first-order scheme is conservative for momentum and total energy, and satisfies a local entropy inequality in its semi-discrete form. The two-dimensional high-order extension is constructed employing the generalized Riemann problem (GRP) in the acoustic approximation. Many numerical tests are presented in order to assess this new scheme. The results obtained for various representative configurations of one and two-dimensional compressible fluid flows show the robustness and the accuracy of our new scheme.

[1]  C. Zemach,et al.  CAVEAT: A computer code for fluid dynamics problems with large distortion and internal slip. Revision 1 , 1992 .

[2]  Pierre-Henri Maire,et al.  Multi-scale Godunov-type method for cell-centered discrete Lagrangian hydrodynamics , 2009, J. Comput. Phys..

[3]  Bruno Després,et al.  Lagrangian Gas Dynamics in Two Dimensions and Lagrangian systems , 2005 .

[4]  John K. Dukowicz,et al.  A general, non-iterative Riemann solver for Godunov's method☆ , 1985 .

[5]  Lev Davidovich Landau,et al.  Mécanique des fluides , 1989 .

[6]  Qiang Zhang,et al.  Small amplitude theory of Richtmyer–Meshkov instability , 1994 .

[7]  Mikhail Shashkov,et al.  Formulations of Artificial Viscosity for Multi-dimensional Shock Wave Computations , 1998 .

[8]  Guglielmo Scovazzi,et al.  Stabilized shock hydrodynamics: II. Design and physical interpretation of the SUPG operator for Lagrangian computations☆ , 2007 .

[9]  Rémi Abgrall,et al.  A Cell-Centered Lagrangian Scheme for Two-Dimensional Compressible Flow Problems , 2007, SIAM J. Sci. Comput..

[10]  Mikhail J. Shashkov,et al.  A Compatible Lagrangian Hydrodynamics Algorithm for Unstructured Grids , 2003 .

[11]  M. Shashkov,et al.  Elimination of Artificial Grid Distortion and Hourglass-Type Motions by Means of Lagrangian Subzonal Masses and Pressures , 1998 .

[12]  J. Falcovitz,et al.  An upwind second-order scheme for compressible duct flows , 1986 .

[13]  A. Bressan.  Hyperbolic Systems of Conservation Laws , 1999 .

[14]  P. Mazur,et al.  Non-equilibrium thermodynamics, , 1963 .

[15]  M. Wilkins.  Calculation of Elastic-Plastic Flow , 1963 .

[16]  B. V. Leer,et al.  Towards the Ultimate Conservative Difference Scheme , 1997 .

[17]  P. Lax.  Hyperbolic systems of conservation laws , 2006 .

[18]  Bruno Després,et al.  A cell-centered Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension , 2009, J. Comput. Phys..

[19]  Mikhail Shashkov,et al.  Multi-Scale Lagrangian Shock Hydrodynamics on Q1/P0 Finite Elements: Theoretical Framework and Two-dimensional Computations. , 2008 .

[20]  R. D. Richtmyer,et al.  A Method for the Numerical Calculation of Hydrodynamic Shocks , 1950 .

[21]  W. F. Noh.  Errors for calculations of strong shocks using an artificial viscosity and artificial heat flux , 1985 .

[22]  M. Berger,et al.  Analysis of Slope Limiters on Irregular Grids , 2005 .

[23]  Raphaël Loubère,et al.  A subcell remapping method on staggered polygonal grids for arbitrary-Lagrangian-Eulerian methods , 2005 .

[24]  M. Shashkov.  Conservative Finite-Difference Methods on General Grids , 1996 .

[25]  Matania Ben-Artzi,et al.  Application of the “generalized Riemann problem” method to 1-D compressible flows with material interfaces , 1986 .

[26]  G. Sod.  A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws , 1978 .

[27]  William J. Rider,et al.  CONSISTENT METRICS FOR CODE VERIFICATION , 2002 .

[28]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[29]  John K. Dukowicz,et al.  Vorticity errors in multidimensional Lagrangian codes , 1992 .

[30]  J. Glinsky,et al.  The general. , 1982, Nursing.

[31]  Thomas J. R. Hughes,et al.  Stabilized shock hydrodynamics: I. A Lagrangian method , 2007 .

[32]  Mikhail Shashkov,et al.  A tensor artificial viscosity using a mimetic finite difference algorithm , 2001 .

[33]  Zhongfeng Sun,et al.  Remark on the generalized Riemann problem method for compressible fluid flows , 2007, J. Comput. Phys..

[34]  Yong Jung Kim.  A MATHEMATICAL INTRODUCTION TO FLUID MECHANICS , 2008 .

[35]  Joseph Falcovitz,et al.  Generalized Riemann Problems in Computational Fluid Dynamics: Bibliography , 2003 .

[36]  Timothy J. Barth,et al.  The design and application of upwind schemes on unstructured meshes , 1989 .

[37]  Blair Swartz,et al.  Good Neighborhoods for Multidimensional Van Leer Limiting , 1999 .

[38]  V. Venkatakrishnan.  Convergence to steady state solutions of the Euler equations on unstructured grids with limiters , 1995 .

[39]  Jérôme Breil,et al.  A second‐order cell‐centered Lagrangian scheme for two‐dimensional compressible flow problems , 2008 .

[40]  R. Kidder,et al.  Laser-driven compression of hollow shells: power requirements and stability limitations , 1976 .

[41]  R. D. Richtmyer.  Taylor instability in shock acceleration of compressible fluids , 1960 .

[42]  M. Shashkov,et al.  The Construction of Compatible Hydrodynamics Algorithms Utilizing Conservation of Total Energy , 1998 .

[43]  J. Falcovitz,et al.  A second-order Godunov-type scheme for compressible fluid dynamics , 1984 .