Abstract
This article considers stabilized finite element and finite volume discretization techniques for systems of conservation laws. Using newly developed techniques in entropy symmetrization theory, simplified forms of the Galerkin least-squares (GLS) and the discontinuous Galerkin (DG) finite element method are developed and analyzed. The use of symmetrization variables yields numerical schemes which inherit global entropy stability properties of the PDE system. Detailed consideration is given to symmetrization of the Euler, Navier-Stokes, and magneto-hydrodynamic (MHD) equations. Numerous calculations are presented to evaluate the spatial accuracy and feature resolution capability of the simplified DG and GLS discretizations. Next, upwind finite volume methods are reviewed. Specifically considered are generalizations of Godunov’s method to high order accuracy and unstructured meshes. An important component of high order accurate Godunov methods is the spatial reconstruction operator. A number of reconstruction operators are reviewed based on Green-Gauss formulas as well as least-squares approximation. Several theoretical results using maximum principle analysis are presented for the upwind finite volume method. To assess the performance of the upwind finite volume technique, various numerical calculations in computational fluid dynamics are provided.
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References
R. Abgrall. An essentially non-oscillatory reconstruction procedure on finite-element type meshes. Comp. Meth. Appl. Mech. Engrg., 116: 95–101, 1994.
R. Abgrall. On essentially non-oscillatory schemes on unstructured meshes. J. Comp. Phys., 114: 45–58, 1995.
P. Arminjon and A. Dervieux. Construction of TVD-like artificial viscosities on two-dimensional arbitrary FEM grids. J. Comp. Phys., 106 (1): 176–198, 1993.
D. Balsara. Higher-order Godunov schemes for isothermal hydrodynamics. Astrophysical J., 420: 197–203, 1994.
T. J. Barth. Some notes on shock resolving flux functions part 1: Stationary characteristics. Technical Report TM-101087, NASA Ames Research Center, Moffett Field, CA, May 1989.
T. J. Barth. Unstructured grids and finite-volume solvers for the Euler and Navier-Stokes equations, March 1991. von Karman Institute Lecture Series 1991–05.
T. J. Barth. Recent developments in high order k-exact reconstruction on unstructured meshes. Technical Report 93–0668, AIAA, Reno, NV, 1993.
T. J. Barth. Aspects of unstructured grids and finite-volume solvers for the Euler and Navier-Stokes equations, March 1994. von Karman Institute Lecture Series 1994–05.
T. J. Barth and D. C. Jespersen. The design and application of upwind schemes on unstructured meshes. Technical Report 89–0366, AIAA, Reno, NV, 1989.
F. Bassi and S. Rebay. High-order accurate discontinuous finite element solution of the 2D Euler equations. J. Comp. Phys., 138 (2): 251–285, 1997.
K. Bey. A Runge-Kutta discontinuous finite element method for high speed flows. Technical Report 91–1575, AIAA, Honolulu, Hawaii, 1991.
M. Brio and C. C. Wu. An upwind differencing scheme for the equations of ideal magnetohydrodynamics. J. Comp. Phys., 75: 400–422, 1988.
G. Chiocchia. Exact solutions to transonic and supersonic flows. Technical Report AR-211, AGARD, 1985.
P. G. Ciarlet and P.-A. Raviart. The combined effect of curved boundaries and numerical integration in isoparametric finite element methods. In A.K. Aziz, editor, The Mathematical Foundations of the Finite Element Method with Application to Partial Differential Equations, pages 409–474, New York, 1972. Academic Press.
P. G. Ciarlet and P.-A. Raviart. Maximum principle and uniform convergence for the finite element method. Comp. Meth. Appl. Mech. Engrg., 2: 17–31, 1973.
B. Cockburn, S. Hou, and C.W. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case. Math. Comp., 54: 545–581, 1990.
B. Cockburn, S.Y. Lin, and C.W. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One dimensional systems. J. Comp. Phys., 84: 90–113, 1989.
B. Cockburn and C.W. Shu. The Runge-Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems. Technical Report 201737, ICASE, NASA Langley R.C., 1997.
P. Collela and P. Woodward. The piecewise parabolic methods for gas-dynamical simulations. J. Comp. Phys., 54, 1984.
M. G. Crandall and A. Majda. Monotone difference approximations for scalar conservation laws. Math. Comp., 34: 1–21, 1980.
B. Delaunay. Sur la sphére vide. Izvestia Akademii Nauk SSSR, 7 (6): 793–800, 1934.
J. Desideri and A. Dervieux. Compressible flow solvers using unstructured grids, March 1988. von Karman Institute Lecture Series 1988–05.
A. C. Galeâo and E. G. Dutra do Carmo. A consistent approximate upwind Petrov-Galerkin method for convection-dominated problems. Comput. Meth. Appl. Mech. Engrg., 68, 1989.
F. R. Gantmacher. Matrix Theory. Chelsea Publishing Company, New York, N.Y., 1959.
S. K. Godunov. A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb., 47, 1959.
S. K. Godunov. An interesting class of quasilinear systems. Dokl. Akad. Nauk. SSSR, 139: 521–523, 1961.
S. K. Godunov. The symmetric form of magnetohydrodynamics equation. Num. Meth. Mech. Cont. Media, 1: 26–34, 1972.
J. D. Goodman and R. J. Le Veque. On the accuracy of stable schemes for 2D conservation laws. Math. Comp., 45, 1985.
A. Harten. High resolution schemes for hyperbolic conservation laws. J. Comp. Phys., 49: 357–393, 1983.
A. Harten. On the symmetric form of systems of conservation laws with entropy. J. Comp. Phys., 49: 151–164, 1983.
A. Harten, J. M. Hyman, and P. D. Lax. On finite-difference approximations and entropy conditions for shocks. Comm. Pure and Appl. Math., 29: 297–322, 1976.
A. Harten, S. Osher, B. Enquist, and S. Chakravarthy. Uniformly high-order accurate essentially nonoscillatory schemes III. J. Comp. Phys.,71(2):231–303, 1987.
T. J. R. Hughes, L. P. Franca, and M. Mallet. A new finite element formulation for CFD: I. symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics. Comp. Meth. Appl. Mech. Engrg., 54: 223–234, 1986.
T. J. R. Hughes and M. Mallet. A new finite element formulation for CFD: III. the generalized streamline operator for multidimensional advective-diffusive systems. Comp. Meth. Appl. Mech. Engrg., 58: 305–328, 1986.
A. Jameson. Analysis and design of numerical schemes for gas dynamics. Technical Report TR 94–15, RIACS, NASA Ames R.C., Moffett Field, CA, 1995.
C. Johnson. Numerical Solution of Partial Differential Equations by the Finite Element Method. Cambridge University Press, Cambridge, 1987.
C. Johnson and J. Pitkäranta. An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comp., 46: 1–26, 1986.
C. Johnson and A. Szepessy. Convergence of the shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws. Math. Comp., 54: 107–129, 1990.
D. Kröner, M. Rokyta, and M. Wierse. A Lax-Wendroff type theorem for upwind finite volume schemes in 2-d. East-West J. Numer. Math.,4(4):279–292, 1996.
D. Kröner, S. Noelle, and M. Rokyta. Convergence of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions. Numer. Math., 71 (4): 527–560, 1995.
C. Lawson. Properties of n-dimensional triangulations. CAGD, 3: 231–246, 1986.
P. D. Lax. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. SIAM, Philadelphia, Penn., 1973.
C. R. Michel. Improved reconstruction schemes for the navier-stokes equations on unstructured meshes. Technical Report 94–0642, AIAA, 1994.
M. S. Mock. Systems of conservation laws of mixed type. J. Diff. Eqns., 37: 70–88, 1980.
K. G. Powell. An approximate Riemann solver for magnetohydrodynamics (that works in more than one dimension). Technical Report 94–24, ICASE, NASA Langley R.C., 1994.
P. L. Roe. Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys., 43, 1981.
P. L. Roe and D. S. Balsam. Notes on the eigensystem of magnetohydrodyamics. SIAM J. Appl. Math., 56 (1): 57–67, 1996.
D. Serre. Remarks about the discrete profiles of shock waves. Mat. Contemp., 11: 153–170, 1996.
F. Shakib. Finite Element Analysis of the Compressible Euler and Navier-Stokes Equations. PhD thesis, Stanford University, Department of Mechanical Engineering, 1988.
S. Spekreijse. Multigrid Solution of the Steady Euler-Equations. PhD thesis, Centrum voor Wiskunde en Informatica, Amsterdam, 1987.
R. Struijs. An adaptive grid polygonal finite volume method for the compressible flow equations. Technical Report 89–1959-CP, AIAA, 1989.
R. Struijs. A Multi-Dimensional Upwind Discretization Method for the Euler Equations on Unstructured Grids. PhD thesis, T.U. Delf and the VKI Institute, 1994.
B. van Leer. Towards the ultimate conservative difference schemes V. a second order sequel to Godunov’s method. J. Comp. Phys., 32, 1979.
P. Vankeirsbilck. Algorithmic Developments for the Solution of Hyperbolic Conservation Laws on Adaptive Unstructured Grids. PhD thesis, Katholiek Universiteit van Leuven, 1993.
V. Venkatakrishnan. On the accuracy of limiters and convergence to steady state. Technical Report 93–0880, AIAA, Reno, NV, 1993.
P. Woodward and P. Colella. The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comp. Phys., 54: 115–173, 1984.
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Barth, T.J. (1999). Numerical Methods for Gasdynamic Systems on Unstructured Meshes. In: Kröner, D., Ohlberger, M., Rohde, C. (eds) An Introduction to Recent Developments in Theory and Numerics for Conservation Laws. Lecture Notes in Computational Science and Engineering, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58535-7_5
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DOI: https://doi.org/10.1007/978-3-642-58535-7_5
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