Abstract
In this paper, a family of sub-cell finite volume schemes for solving the hyperbolic conservation laws is proposed and analyzed in one-dimensional cases. The basic idea of this method is to subdivide a control volume (main cell) into several sub-cells and the finite volume discretization is applied to each of the sub-cells. The averaged values on the sub-cells of current and face neighboring main cells are used to reconstruct the polynomial distributions of the dependent variables. This method can achieve arbitrarily high order of accuracy using a compact stencil. It is similar to the spectral volume method incorporating with PNPM technique but with fundamental differences. An elaborate utilization of these differences overcomes some shortcomings of the spectral volume method and results in a family of accurate and robust schemes for solving the hyperbolic conservation laws. In this paper, the basic formulation of the proposed method is presented. The Fourier analysis is performed to study the properties of the one-dimensional schemes. A WENO limiter based on the secondary reconstruction is constructed.
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References
T. J. Barth, and P. O. Frederickson, "Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction", AIAA Paper No. 90 0013 (1990).
M. Delanaye, "Quadratic reconstruction finite volume schemes on 3D arbitrary unstrtictured polyhedral grids", AIAA Paper No. 99 3295 (1989).
C. Ollivier-Gooch, and M. Van Altena, J. Comp. Phys. 181 729 (2002).
C. F. Ollivier-Gooch, J. Comp. Phys. 133 6 (1997).
R. Abgrall, J. Comp. Phys. 114 45 (1994).
O. Friedrich, J. Comp. Phys. 144 194 (1998).
M. Dumbser, and M. Kaser, J. Comp. Phys. 221 693 (2007).
M. Dumbser, M. Kaser, V. A. Titarev, and E. F. Toro, J. Comp. Phys. 226 204 (2007).
C. Hu, and C. W. Shu, J. Comp. Phys. 150 97 (1999).
W. Li, and Y. X. Ren, Int. J. Numer. Meth. Fluids 70 742 (2012).
W. Li, and Y. X. Ren, Comp. Fluids 96 368 (2014).
W. H. Reed, and T. Hill, Triangular mesh methods for the neutron transport equation, Los Alamos Report (LA-UR-73-479).
B. Cockburn, and C. W. Shu, Math. Comput. 52 411 (1989).
B. Cockburn, S. Y. Lin, and C. W. Shu, J. Comp. Phys. 84 90 (1989).
B. Cockburn, S. Hou, and C. W. Shu, Math. Comput. 54 545 (1990).
B. Cockburn, and C. W. Shu, SIAM J. Numer. Anal. 35 2440 (1998).
F. Bassi, and S. Rebay, J. Comp. Phys. 131 267 (1997).
Z. J. Wang, J. Comp. Phys. 178 210 (2002).
Z. J. Wang, and Y. Liu, J. Comp. Phys. 179 665 (2002).
Z. J. Wang, and Y. Liu, J. Sci. Comput. 20 137 (2004).
Z. J. Wang, L. Zhang, and Y. Liu, J. Comp. Phys. 194 716 (2004).
M. Dumbser, Comput. Fluids 39 60 (2010).
M. Dumbser, and O. Zanotti, J. Comp. Phys. 228 6991 (2009).
L. Zhang, L. W. Liu, L. X. He, X. G. Deng, and H. X. Zhang, J. Comp. Phys. 231 1081 (2012).
L. Zhang, W. Liu, L. X. He, X. G. Deng, and H. X. Zhang, J. Comp. Phys. 231 1104 (2012).
L. Zhang, W. Liu, L. He, and X. Deng, Commun. Commut. Phys. 12 284 (2012).
L. Zhang, W. Liu, M. Li, X. He, and H. Zhang, Comput. Fluids 97 110 (2014).
H. Luo, L. Luo, R. Nourgaliev, V. A. Mousseau, and N. Dinh, J. Comp. Phys. 229 6961 (2010).
D. W. Zingg, S. De Rango, M. Nemec, and T. H. Pulliam, J. Comp. Phys. 160 683 (2000).
J. Pan, and Y. Ren, in High order sub-cell finite volume method in solving hyperbolic conservation laws: Proceedings of 22nd A/AA Computational Fluid Dynamics Conference (American Institute of Aeronautics and Astronautics, Dallas, 2015), p. 2286.
J. Pan, Y. Ren, and Y. Sun, J. Comp. Phys. 338 165 (2017).
P. L. Roe, J. Comp. Phys. 43 357 (1981).
S. Gottlieb, C. W. Shu, and E. Tadmor, SIAM Rev. 43 89 (2001).
F. Q. Hu, M. Y. Hussaini, and P. Rasetarinera, J. Comp. Phys. 151 921 (1999).
S. K. Lele, J. Comp. Phys. 103 16 (1992).
K. Van den Abeele, T. Broeckhoven, and C. Lacor, J. Comp. Phys. 224 616 (2007).
C. W. Shu, Essentially Non-Oscillatory and Weighted Essentially NonOscillatory Schemes for Hyperbolic Conservation Laws (Springer, Berlin, Heidelberg, 1998).
G. S. Jiang, and C. W. Shu, J. Comput. Phys. 126 202 (1996).
G. A. Sod, J. Comp. Phys. 27 1 (1978).
P. D. Lax, Comm. Pure Appl. Math. 7 159 (1954).
P. Woodward, and P. Colella, J. Comp. Phys. 54 115 (1984).
C. W. Shu, and S. Osher, J. Comp. Phys. 77 439 (1988).
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Pan, J., Ren, Y. High order sub-cell finite volume schemes for solving hyperbolic conservation laws I: basic formulation and one-dimensional analysis. Sci. China Phys. Mech. Astron. 60, 084711 (2017). https://doi.org/10.1007/s11433-017-9033-9
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DOI: https://doi.org/10.1007/s11433-017-9033-9