An adaptive-order discontinuous Galerkin method for the solution of the Euler equations of gas dynamics

International Journal for Numerical Methods in Engineering

Volumn 47, Issue 1-3, 2000, Pages 61-73

  Baumann, Carlos Erik ^a,b   Oden, J Tinsley ^a,c  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^b Taylor Hall 2400 ^*  (United States)

^c TICAM   (United States)

Author keywords

Discontinuous finite elements; Euler equations; Galerkin methods; Gas dynamics

Indexed keywords

APPROXIMATION THEORY; BOUNDARY CONDITIONS; CONVERGENCE OF NUMERICAL METHODS; FINITE ELEMENT METHOD; POLYNOMIALS; PROBLEM SOLVING;

ADAPTIVE-ORDER DISCONTINUOUS GALERKIN METHODS; EULER EQUATIONS;

GALERKIN METHODS;

DYNAMIC MODELING; FINITE ELEMENT ANALYSIS; GAS FLOW;

EID: 0034628097     PISSN: 00295981     EISSN: None     Source Type: Journal    
DOI: 10.1002/(sici)1097-0207(20000110/30)47:1/3<61::aid-nme761>3.0.co;2-d     Document Type: Article

References (31)

1. Roe PL, Error estimates for cell-vertex solutions of the compressible Euler equations, Technical Report 87-6, ICASE, January 1987.
2. Turkel E. Accuracy of schemes with nonuniform meshes for compressible fluid flows. Technical Report 85-43, ICASE, September 1985.
3. Harten A, Osher S. Uniformly high-order accurate nonoscillatore schemes. SIAM Journal of Numerecal Analysis 1987; 24:279-309.
4. Van Leer B. Towards the ultimate conservative difference scheme. III. upstresm-centered finite difference schemes for ideal compressible flow. Journal of Computational Physics 1977: 23:263-175.
5. Woodward PR, Colella P. The piecewise parabolic method (PPM) for gas dynamical calculations, Journal of Computational Phycics 1984; 54:174-201.
6. Lesaint P, Raviart PA. On a finite element method for solving the neutron transport equation. In Mathematical Aspects of Finite Elements in Partial Defferential Equations, de Boor C (ed.). Academic Press: New York, 1974; 89-123.
7. Johnson C, Pitkäranta J. An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Mathematics of Computation 1986; 46:1-26.
8. Bey KS. An hp-adaptive discontinuous Galerkin method of hyperbolic conservation laws. Ph.D. Dissertation, The University of Texas at Austin, May 1994.
9. Cockburn B, Hou S, Shu CW. TVB Runge-Kutta local projection discontinuious Galerkin finite element for conservation laws IV: the milti-dimensional case. Mathematics of Computation 1990; 54:545.
10. Cockburn B. Hou S, Shy CW. The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, ICASE Report 97-43, 1997.
11. Allmaras SR. A coupled Euler/Navier-Stokes algorithm for 2-D unsteady transonic shock/boundary-layer interaction. Ph.D. Dissertation, Massachusetts Institute of Technology, February 1989.
12. Lowrie RB. Compact Higher-Order Numerical Methods for Hyperbolic Conservation Laws. ph.D. Dissertation, University of Michigan, 1996.
13. Bey KS, Oden JT. A Runge-Kutta discontinuous finite element for high speed flows, In AIAA 10th Computational Fluid Dynamics Conference, June 1991.
14. Baumann CE, Oden JT. A discontinuous hp finite element method for the solution of the Euler equations of Gas Dynamics. In Tenth International Conference on Finite Elements in Fluids, 5-8 January 1998.
15. Hu C. Shu C-W. A discontinuous Galerkin finite element method for the Hamilton-Jacobi equation, Division of Applied Mathematics, Brown University, 1997.
16. Atkins HL, Shu C-W. Quadrature-free implementation of discontinuous Galerkin methods for hyperbolic equations. ICASE Report 96-51, 1996.
17. Bassi F, Rebay R, Savini M, Pedinotti S. The Discontiuuous Galerkin method applied to CFD problems. In Second European Conference on Turbomachinery, Fluid Dynamics and Thermodynamics. ASME: New york, 1995.
18. Lomtev I, Kaniadakis GE, A discontiuous Galerkin method for the Navier-Stokes equations, International Journal for Numerical Methods in Fluids 1999; 29:587-603.
19. Lomtev I, Quillen CB, Karniadakis GE, Spectral/hp methods for viscous compressible flows on unstructural 2nd meshes. Journal of Computational Physics 1998, to appear.
20. Warburton TC, Lomtev I, Kirby RM, Karniadakis GE, A discontinuous Galerkin method for the Navier-Stokes equations on hybrid grids. Center for Fluid Mechanics 97-14, Division of Applied Mathematics, Brown Universisty, 1997.
21. Cockburn B. An introduction to the discontiuous Galerkin method for convection-dominated problems. School of Mathematics, University of Minnesota, 1997.
22. Baumann CE, Oden JT. A discontiuous hp finite element method for convection-diffusion problems. TICAM Report 97-23. Computer Methods in Applied Mechanics and Engineering 1999; 175:311-341.
23. Hirsch C. Numerical Computation of Internal and External Flows. Wiley: New York, 1990.
24. Osher S, Chakravarthy S. Upwind schemes and boundary conditions with applications to Euler equations in general geometries. Jounal of Computational Physics 1983; 50:447-481.
25. Baumann CE. An hp-adaptrve discontinuous finite element method for computational fluid dynamics. Ph.D. Dissertation, The University of Texas at Austin, August 1997.
26. Van Leer B. Flux-vector splitting for the Euler equations. Lecture Notes in Physics 1982; 170:507-512.
27. Roe PL. Approximate Riemann solvers, parameter vectors and difference schemes. Journal of Computational Physics 1981; 43:357-372.
28. Steger JL, Warming RF. Flux vector splitting of the inviscid gas dynamics equations with applications to finite difference methods. Journal of Computational Physics 1981; 40:263-293.
29. Dye WH, Polek T. Results of pressure distribution tests on a 0.010-scale space shuttle orbiter (61-0) in the nasa/arc 3.5-foot hypersonic wind tunnel (test oh38). Technical Report, NASA, 1975
30. Oden JT, Babuška I, Baumann CE. A discontinuous hp finite element method for diffusion problems. Journal of Computational Physics 1998; 146:1-29. also TICAM Report 97-21.
31. Oden JT, Babuška I, Baumann CE. A discontinuous hp finite element method for diffusion problems. Journal of Computational Physics 1998; 146:1-29. also TICAM Report 97-21.