Efficient quadrature-free high-order spectral volume method on unstructured grids: Theory and 2D implementation

Journal of Computational Physics

Volumn 227, Issue 3, 2008, Pages 1620-1642

  Harris, Rob ^a   Wang, Z J ^a   Liu, Yen ^b  

^a Iowa State University of Science and Technology   (United States)

^b NASA AMES RESEARCH CENTER   (United States)

Author keywords

Euler equations; Hyperbolic conservation laws; Spectral volume method; Unstructured grid

Indexed keywords

EFFICIENCY; GAUSSIAN DISTRIBUTION; PHYSICAL PROPERTIES; POLYNOMIAL APPROXIMATION;

CURVED BOUNDARY; EULER'S EQUATION; GAUSS QUADRATURE FORMULA; GRID THEORY; HIGH-ORDER; HIGHER-ORDER; HYPERBOLIC CONSERVATION LAWS; SPECTRAL VOLUME METHOD; SPECTRAL VOLUMES; UNSTRUCTURED GRID;

EULER EQUATIONS;

EID: 37249066633     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2007.09.012     Document Type: Article

References (39)

1. Abgrall R. On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation. J. Comput. Phys. 114 (1994) 45-58
2. Atkins H.L., and Shu C.-W. Quadrature-free implementation of the discontinuous Galerkin method for hyperbolic equations. AIAA J. 96 (1996) 1683
3. T.J. Barth, P.O. Frederickson, High-order solution of the Euler equations on unstructured grids using quadratic reconstruction, AIAA Paper No. 90-0013, 1990.
4. Bassi F., and Rebay S. High-order accurate discontinuous finite element solution of the 2D Euler equations. J. Comput. Phys. 138 (1997) 251-285
5. Batten P., Clarke N., Lambert C., and Causon D.M. On the choice of wavespeeds for the HLLC Riemann solver. SIAM J. Sci. Comput. 18 6 (1997) 1553-1570
6. Chen Q.Y. Partitions for spectral (finite) volume reconstruction in the tetrahedron. SIAM J. Sci. Comput. (2005)
7. Cockburn B., and Shu C.-W. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52 (1989) 411-435
8. Cockburn B., Lin S.-Y., and Shu C.-W. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems. J. Comput. Phys. 84 (1989) 90-113
9. Cockburn B., Hou S., and Shu C.-W. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Math. Comput. 54 (1990) 545-581
10. Cockburn B., and Shu C.-W. The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141 (1998) 199-224
11. M. Delanaye, Yen Liu, Quadratic reconstruction finite volume schemes on 3D arbitrary unstructured polyhedral grids, AIAA Paper No. 99-3259-CP, 1999.
12. Godunov S.K. A finite-difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb. 47 (1959) 271
13. Gottlieb S., Shu C.-W., and Tadmor E. Strong stability-preserving high-order time discretization methods. SIAM Rev. 43 1 (2001) 89-112
14. Harten A., Engquist B., Osher S., and Chakravarthy S. Uniformly high order essentially non-oscillatory schemes III. J. Comput. Phys. 71 (1987) 231
15. Harten A., Lax P.D., and Van Leer B. On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25 (1983) 35-61
16. Hesthaven J.S. From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex. SIAM J. Numer. Anal. 35 2 (1998) 655-676
17. Hesthaven J.S., and Teng C.H. Stable spectral methods on tetrahedral elements. SIAM J. Sci. Comput. 21 6 (2000) 2352-2380
18. Hu C., and Shu C.-W. Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 150 (1999) 97-127
19. Krivodonova L., and Berger M. High-order accurate implementation of solid wall boundary conditions in curved geometries. J. Comput. Phys. 211 (2006) 492-512
20. Liu Y., Vinokur M., and Wang Z.J. Spectral (finite) volume method for conservation laws on unstructured grids V: extension to three-dimensional systems. J. Comput. Phys. 212 (2006) 454-472
21. H. Luo, J. Baum, R. Löhner, On the computation of steady-state compressible flows using a discontinuous Galerkin method, presented at the Fourth International Conference on Computational Fluid Dynamics, July 10-14, Ghent, Belgium, 2006.
22. Roe P.L. Approximate Riemann solvers, parameter vectors and difference schemes. J. Comput. Phys. 43 (1981) 357-372
23. Rusanov V.V. Calculation of interaction of non-steady shock waves with obstacles. J. Comput. Math. Phys. USSR 1 (1961) 267-279
24. Sun Y., Wang Z.J., and Liu Y. High-order multidomain spectral difference method for the Navier-Stokes equations on unstructured hexahedral grids. Commun. Comput. Phys. 2 (2007) 310-333
25. Sun Y., Wang Z.J., and Liu Y. Spectral (finite) volume method for conservation laws on unstructured grids VI: extension to viscous flow. J. Comput. Phys. 215 (2006) 41-58
26. Toro E.F., Spruce M., and Speares W. Restoration of the contact surface in the HLL Riemann solver. Shock Waves 4 (1994) 25-34
27. Tu S., and Aliabadi S. A slope limiting procedure in discontinuous Galerkin finite element method for gasdynamics applications. Int. J. Numer. Anal. Mod. 2 2 (2005) 163-178
28. van Leer B. Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method. J. Comput. Phys. 32 (1979) 101-136
29. van Leer B. Upwind and high-resolution methods for compressible flow: From donor cell to residual-distribution schemes. Commun. Comput. Phys. 1 (2006) 192-206
30. Wang Z.J. Spectral (finite) volume method for conservation laws on unstructured grids: basic formulation. J. Comput. Phys. 178 (2002) 210
31. Wang Z.J., and Liu Y. Spectral (finite) volume method for conservation laws on unstructured grids II: extension to two-dimensional scalar equation. J. Comput. Phys. 179 (2002) 665
32. Wang Z.J., and Liu Y. Spectral (finite) volume method for conservation laws on unstructured grids III: extension to one-dimensional systems. J. Sci. Comput. 20 (2004) 137
33. Wang Z.J., and Liu Y. Spectral (finite) volume method for conservation laws on unstructured grids IV: extension to two-dimensional Euler equations. J. Comput. Phys. 194 (2004) 716
34. Wang Z.J., and Liu Y. Extension of the spectral volume method to high-order boundary representation. J. Comput. Phys. 211 (2006) 154-178
35. Wolfram S. Mathematica Book. fourth ed. (1999), Wolfram Media and Cambridge University Press, New York
36. Woodward P., and Colella P. The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54 (1984) 115-173
37. Xing Y., and Shu C.W. A new approach of high order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms. Commun. Comput. Phys. 1 (2006) 100-134
38. Zalesak S.T. Fully multidimensional flux-corrected transport algorithms for fluids. J. Comput. Phys. 32 (1979) 335-362
39. Zhang M., and Shu C.W. An analysis and a comparison between the discontinuous Galerkin method and the spectral finite volume methods. Comput. Fluids 34 4-5 (2005) 581-592