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Entropy stability and well-balancedness of space-time DG for the shallow water equations with bottom topography
1. | Seminar for Applied Mathematics (SAM), Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland, Switzerland |
References:
[1] |
E. Audusse, F. Bouchut, M. O. Bristeau, R. Klien and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM. J. Sci. Comp., 25 (2004), 2050-2065.
doi: 10.1137/S1064827503431090. |
[2] |
M. Castro, J. M. Gallardo and C. Parés, High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with non-conservative products, Math. Comp., 75 (2006), 1103-1134.
doi: 10.1090/S0025-5718-06-01851-5. |
[3] |
C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer, Berlin, 2000.
doi: 10.1007/3-540-29089-3_14. |
[4] |
U. S. Fjordholm, S. Mishra and E. Tadmor, Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography, Journal of Computational Physics, 230 (2011), 5587-5609.
doi: 10.1016/j.jcp.2011.03.042. |
[5] |
J. M. Greenberg and A. Y. LeRoux, A well-balanced scheme for numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal., 33 (1996), 1-16.
doi: 10.1137/0733001. |
[6] |
A. Hiltebrand, Entropy-stable Discontinuous Galerkin Finite Element Methods with Streamline Diffusion and Shock-capturing for Hyperbolic Systems of Conservation Laws, Ph.D thesis, ETH Zurich, 2014, No. 22279. |
[7] |
A. Hiltebrand and S. Mishra, Entropy stable shock capturing space-time discontinuous Galerkin schemes for systems of conservation laws, Numerische Mathematik, 126 (2014), 103-151.
doi: 10.1007/s00211-013-0558-0. |
[8] |
J. Jaffre, C. Johnson and A. Szepessy, Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws, Math. Model. Meth. Appl. Sci., 5 (1995), 367-386.
doi: 10.1142/S021820259500022X. |
[9] |
S. Jin, A steady state capturing method for hyperbolic systems with geometrical source terms, Math. Model. Numer. Anal., 35 (2001), 631-645.
doi: 10.1051/m2an:2001130. |
[10] |
S. Jin and X. Wen, An efficient method for computing hyperbolic systems with geometrical source terms having concentrations, J. Comput. Math., 22 (2004), 230-249. |
[11] |
A. Kurganov and D. Levy, Central-upwind schemes for the St. Vernant system, Math. Model. Num. Anal., 36 (2002), 397-425.
doi: 10.1051/m2an:2002019. |
[12] |
R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002.
doi: 10.1017/CBO9780511791253. |
[13] |
R. J. LeVeque, Balancing source terms and flux gradients in high-resolution Godunov methods: The quasi-steady wave-propagation algorithm, J. Comput. Phys., 146 (1998), 346-365.
doi: 10.1006/jcph.1998.6058. |
[14] |
S. Noelle, N. Pankratz, G. Puppo and J. Natvig, Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows, J. Comput. Phys., 213 (2006), 474-499.
doi: 10.1016/j.jcp.2005.08.019. |
show all references
References:
[1] |
E. Audusse, F. Bouchut, M. O. Bristeau, R. Klien and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM. J. Sci. Comp., 25 (2004), 2050-2065.
doi: 10.1137/S1064827503431090. |
[2] |
M. Castro, J. M. Gallardo and C. Parés, High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with non-conservative products, Math. Comp., 75 (2006), 1103-1134.
doi: 10.1090/S0025-5718-06-01851-5. |
[3] |
C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer, Berlin, 2000.
doi: 10.1007/3-540-29089-3_14. |
[4] |
U. S. Fjordholm, S. Mishra and E. Tadmor, Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography, Journal of Computational Physics, 230 (2011), 5587-5609.
doi: 10.1016/j.jcp.2011.03.042. |
[5] |
J. M. Greenberg and A. Y. LeRoux, A well-balanced scheme for numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal., 33 (1996), 1-16.
doi: 10.1137/0733001. |
[6] |
A. Hiltebrand, Entropy-stable Discontinuous Galerkin Finite Element Methods with Streamline Diffusion and Shock-capturing for Hyperbolic Systems of Conservation Laws, Ph.D thesis, ETH Zurich, 2014, No. 22279. |
[7] |
A. Hiltebrand and S. Mishra, Entropy stable shock capturing space-time discontinuous Galerkin schemes for systems of conservation laws, Numerische Mathematik, 126 (2014), 103-151.
doi: 10.1007/s00211-013-0558-0. |
[8] |
J. Jaffre, C. Johnson and A. Szepessy, Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws, Math. Model. Meth. Appl. Sci., 5 (1995), 367-386.
doi: 10.1142/S021820259500022X. |
[9] |
S. Jin, A steady state capturing method for hyperbolic systems with geometrical source terms, Math. Model. Numer. Anal., 35 (2001), 631-645.
doi: 10.1051/m2an:2001130. |
[10] |
S. Jin and X. Wen, An efficient method for computing hyperbolic systems with geometrical source terms having concentrations, J. Comput. Math., 22 (2004), 230-249. |
[11] |
A. Kurganov and D. Levy, Central-upwind schemes for the St. Vernant system, Math. Model. Num. Anal., 36 (2002), 397-425.
doi: 10.1051/m2an:2002019. |
[12] |
R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002.
doi: 10.1017/CBO9780511791253. |
[13] |
R. J. LeVeque, Balancing source terms and flux gradients in high-resolution Godunov methods: The quasi-steady wave-propagation algorithm, J. Comput. Phys., 146 (1998), 346-365.
doi: 10.1006/jcph.1998.6058. |
[14] |
S. Noelle, N. Pankratz, G. Puppo and J. Natvig, Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows, J. Comput. Phys., 213 (2006), 474-499.
doi: 10.1016/j.jcp.2005.08.019. |
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