Construction of stabilization operators for Galerkin least-squares discretizations of compressible and incompressible flows

The design and analysis of a class of stabilization operators suitable for space-time Galerkin least-squares finite element discretizations of the symmetrized compressible Navier-Stokes equations is discussed. The obtained stabilization matrix is well defined in the incompressible limit and reduces to the matrix described in [M. Polner, J.J.W. van der Vegt and R.M.J. van Damme, Analysis of stabilization operators for Galerkin least-squares discretizations of the incompressible Navier-Stokes equations, Comput. Meth. Appl. Mech. Eng., 195(2006):982--1006]. The stabilization matrix is investigated to give necessary and sufficient conditions for its positive definiteness. Under these conditions on the stabilization matrix, the Galerkin least-squares method for the symmetrized compressible Navier-Stokes equations satisfies the entropy condition, which ensures global nonlinear stability of the discretization.

[1]  G. Kell.  Density, thermal expansivity, and compressibility of liquid water from 0.deg. to 150.deg.. Correlations and tables for atmospheric pressure and saturation reviewed and expressed on 1968 temperature scale , 1975 .

[2]  U. Ghia,et al.  High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method , 1982 .

[3]  David L. Darmofal,et al.  The solution of the compressible Euler equations at low Mach numbers using a stabilized finite element algorithm , 2001 .

[4]  Peter Hansbo,et al.  On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws , 1990 .

[5]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuscka-Brezzi condition: A stable Petrov-Galerkin formulation of , 1986 .

[6]  L. Franca,et al.  Stabilized finite element methods. II: The incompressible Navier-Stokes equations , 1992 .

[7]  A. Harten.  On the symmetric form of systems of conservation laws with entropy , 1983 .

[8]  Thomas J. R. Hughes,et al.  A comparative study of different sets of variables for solving compressible and incompressible flows , 1998 .

[9]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: II. Beyond SUPG , 1986 .

[10]  Jim Douglas,et al.  An absolutely stabilized finite element method for the stokes problem , 1989 .

[11]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics. X - The compressible Euler and Navier-Stokes equations , 1991 .

[12]  Mónika Polner,et al.  Galerkin Least-Squares Stabilization Operators for the Navier-Stokes Equations: A Unified Approach , 2005 .

[13]  S. K. Godunov,et al.  THE PROBLEM OF A GENERALIZED SOLUTION IN THE THEORY OF QUASILINEAR EQUATIONS AND IN GAS DYNAMICS , 1962 .

[14]  H. Schlichting.  Boundary Layer Theory , 1955 .

[15]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier—Stokes equations and the second law of thermodynamics , 1986 .

[16]  F. W. Sears,et al.  Thermodynamics, kinetic theory, and statistical thermodynamics , 1975 .

[17]  T. Hughes,et al.  Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations , 1990 .

[18]  Guillermo Hauke,et al.  a Unified Approach to Compressible and Incompressible Flows and a New Entropy-Consistent Formulation of the K - Model. , 1994 .

[19]  Thomas J. R. Hughes,et al.  A new finite element formulation for computational fluid dynamics: III. The generalized streamline operator for multidimensional advective-diffusive systems , 1986 .

[20]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[21]  Franco Brezzi,et al.  $b=\int g$ , 1997 .

[22]  T. Hughes,et al.  The Galerkin/least-squares method for advective-diffusive equations , 1988 .

[23]  Timothy J. Barth,et al.  Numerical Methods for Gasdynamic Systems on Unstructured Meshes , 1997, Theory and Numerics for Conservation Laws.

[24]  M. Mock,et al.  Systems of conservation laws of mixed type , 1980 .

[25]  Analysis of stabilization operators for Galerkin least-squares discretizations of the incompressible Navier-Stokes equations , 2006 .

[26]  Thomas J. R. Hughes,et al.  Symmetrization of conservation laws with entropy for high-temperature hypersonic computations , 1990 .

[27]  P. Dutt,et al.  Stable boundary conditions and difference schemas for Navier-Stokes equations , 1988 .