Lagrangian Gas Dynamics in Two Dimensions and Lagrangian systems

We propose a new and canonical way of writing the equations of gas dynamics in Lagrangian coordinates in two dimensions as a weakly hyperbolic system of conservation laws. One part of the system is called the physical part and contains physical variables; the other part is the geometrical part. We show that the physical part is symmetrizable. We show that the weak hyperbolicity is due to shear contact discontinuities. Free divergence constraints play an important role in the system. We prove the L2 stability of the physical part of the system. Based on this formulation, we derive a new conservative and entropy-consistent finite-volume numerical scheme. We prove the stability of the numerical scheme. Numerical results show the potential interest of this approach. Various examples (Born-Infeld, MHD, 3D lagrangian gas dynamics) can be written using the same abstract formalism.

[1]  F. Bouchut.  Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws: and Well-Balanced Schemes for Sources , 2005 .

[2]  S. K. Godounov.  Lois de conservation et integrales d'energie des equations hyperboliques , 1987 .

[3]  Max Born,et al.  Foundations of the new field theory , 1934 .

[4]  D. Benson.  Computational methods in Lagrangian and Eulerian hydrocodes , 1992 .

[5]  B. François,et al.  Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness , 1999 .

[6]  Yann Brenier,et al.  Hydrodynamic Structure of the Augmented Born-Infeld Equations , 2004 .

[7]  Guy Boillat,et al.  Non linear hyperbolic fields and waves , 1996 .

[8]  R. D. Richtmyer,et al.  Difference methods for initial-value problems , 1959 .

[9]  Bruno Després,et al.  Lagrangian systems of conservation laws , 2001, Numerische Mathematik.

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

[11]  Ilio Galligani,et al.  Mathematical Aspects of Finite Element Methods , 1977 .

[12]  A new Lagrangian method for steady supersonic flow computation I. Godunov scheme , 1990 .

[13]  P. Raviart,et al.  Numerical Approximation of Hyperbolic Systems of Conservation Laws , 1996, Applied Mathematical Sciences.

[14]  B. Perthame,et al.  Relaxation of Energy and Approximate Riemann Solvers for General Pressure Laws in Fluid Dynamics , 1998 .

[15]  Structure des systèmes de lois de conservation en variables lagrangiennes , 1999 .

[16]  H. Kreiss,et al.  Initial-Boundary Value Problems and the Navier-Stokes Equations , 2004 .

[17]  Hyperbolicity of the Nonlinear Models of Maxwell’s Equations , 2004 .

[18]  David H. Wagner,et al.  Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions , 1987 .

[19]  A. Ramage,et al.  Computational solution of two-dimensional unsteady PDEs using moving mesh methods , 2002 .

[20]  P. Raviart,et al.  A mixed finite element method for 2-nd order elliptic problems , 1977 .

[21]  Y. Brenier,et al.  Sticky Particles and Scalar Conservation Laws , 1998 .

[22]  Sophia Demoulini,et al.  A Variational Approximation Scheme for¶Three-Dimensional Elastodynamics¶with Polyconvex Energy , 2001 .

[23]  C. Dafermos.  Hyberbolic Conservation Laws in Continuum Physics , 2000 .

[24]  Rémi Abgrall,et al.  A Lagrangian Discontinuous Galerkin‐type method on unstructured meshes to solve hydrodynamics problems , 2004 .

[25]  Wai How Hui,et al.  A unified coordinate system for solving the three-dimensional Euler equations , 1999 .

[26]  Zi-niu Wu.  A Note on the Unified Coordinate System for Computing Shock Waves , 2002 .

[27]  Qin Tiehu,et al.  Symmetrizing Nonlinear Elastodynamic System , 1998 .