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An adaptive discretization of shallow-water equations based on discontinuous Galerkin methods. (English) Zbl 1106.76044
Summary: We present a discontinuous Galerkin formulation of the shallow-water equations. An orthogonal basis is used for spatial discretization, and an explicit Runge-Kutta scheme is used for time discretization. Some results of second-order anisotropic adaptive calculations are presented for dam breaking problems. The adaptive procedure uses an error indicator that concentrates the computational effort near discontinuities like hydraulic jumps.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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[1] . Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973.
[2] , (eds). Discontinuous Galerkin Methods, Lecture Notes in Computational Science and Engineering, vol. 11. Springer: Berlin, 2000.
[3] . A discontinuous Galerkin method for the shallow water equations with source terms. In Discontinuous Galerkin Methods, , (eds), Lecture Notes in Computational Science and Engineering, vol. 11. Springer: Berlin, 2000; 419–424. · Zbl 1041.76512
[4] Harten, SIAM Review 25 pp 35– (1983)
[5] Giraldo, Journal of Computational Physics 181 pp 499– (2002)
[6] Aizinger, Advances in Water Resources 25 pp 67– (2002) · Zbl 1179.76049
[7] Cockburn, Mathematics of Computations 54 pp 545– (1990)
[8] Furtado, Front Tracking and the Interaction of Nonlinear Hyperbolic Waves 43 pp 99– (1989)
[9] Remacle, SIAM Review 45 pp 53– (2003)
[10] Dubiner, Journal of Scientific Computation 6 pp 345– (1991) · Zbl 0735.15006
[11] Yen, Journal of Hydraulic Engineering 118 pp 1326– (1992)
[12] Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction. Springer: Berlin, Heidelberg, 1999.
[13] Numerical Methods for Conservation Laws. Birkhäuser: Basel, 1992. · Zbl 0847.65053
[14] Godunov, Mathematics of the Sbornik 47 pp 207– (1959)
[15] Woodward, Journal of Computational Physics 54 pp 115– (1984)
[16] Roe, Journal of Computational Physics 43 pp 357– (1981)
[17] Glaister, Journal of Hydraulic Research 3 pp 293– (1988)
[18] Alcrudo, International Journal for Numerical Methods in Fluids 14 pp 1009– (1992)
[19] Soares Frazão, Journal of Hydraulic Engineering, American Society of Civil Engineers 128 pp 956– (2002)
[20] Adjerid, Computer Methods in Applied Mechanics and Engineering 191 pp 1097– (2002)
[21] Remacle, International Journal for Numerical Methods in Engineering 58 pp 349– (2003)
[22] Krivodonovna, Applied Numerical Mathematics 48 pp 323– (2004)
[23] Mesh Modification procedures for general 3-D non-manifold domains. Ph.D. Thesis, Rensselear Polytechnic Institute, August 2003.
[24] Finite Volume Methods for Hyperbolic Problems. Cambridge University Press: Cambridge, 2002. · Zbl 1010.65040
[25] Soares Frazão, Journal of Hydraulic Research
[26] Capart, Experiments in Fluids 32 pp 121– (2002)
[27] Spinewine, Experiments in Fluids 34 pp 227– (2003)
[28] Cockburn, Mathematics of Computation 52 pp 411– (1989)
[29] van Leer, Journal of Computational Physics 14 pp 361– (1974)
[30] van Leer, Journal of Computational Physics 30 pp 1– (1979)
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