Abstract
Free boundaries in shallow-water equations demarcate the time-dependent water line between ‘‘flooded’’ and ‘‘dry’’ regions. We present a novel numerical algorithm to treat flooding and drying in a formally second-order explicit space discontinuous Galerkin finite-element discretization of the one-dimensional or symmetric shallow-water equations. The algorithm uses fixed Eulerian flooded elements and a mixed Eulerian–Lagrangian element at each free boundary. When the time step is suitably restricted, we show that the mean water depth is positive. This time-step restriction is based on an analysis of the discretized continuity equation while using the HLLC flux. The algorithm and its implementation are tested in comparison with a large and relevant suite of known exact solutions. The essence of the flooding and drying algorithm pivots around the analysis of a continuity equation with a fluid velocity and a pseudodensity (in the shallow water case the depth). It therefore also applies, for example, to space discontinuous Galerkin finite-element discretizations of the compressible Euler equations in which vacuum regions emerge, in analogy of the above dry regions. We believe that the approach presented can be extended to finite-volume discretizations with similar mean level and slope reconstruction.
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E. Audusse M.-O. Bristeau (2003) ArticleTitleTransport of pollutant in shallow water: a two time steps kinetic method ESAIM-Math. Mod. Numer. Analysis 37 389–416 Occurrence Handle10.1051/m2an:2003034
P. Batten N. Clarke C. Lambert D. M. Causon (1997) ArticleTitleOn the choice of wavespeeds for the HLLC Riemann solver SIAM. J. Sci. Comput. 18 1553–1570 Occurrence Handle10.1137/S1064827593260140
Bokhove O. (2003). Flooding and drying in Finite-Element Discretizations of Shallow-Water Equations. Part 2: Two Dimensions. Memorandum 1684, Mathematical Communications 2003 University of Twente, www.math.utwente.nl/publications/
Bokhove, O., and Wirosoetisno, D. (2003). Drying and wetting in finite element shallow-water flows. In Jirka, G. H. and Uijttewaal, W. S. J. (eds.), {\em Proceedings of the International Symposium on Shallow Flows}, Delft University of Technology, 16–18 June 2003, Part III, pp. 153–160.
Bokhove, O., Woods, A. W., and Boer de, A. (2004). Magma Flow through Elastic-Walled Dikes. Submitted to Theoretical and Numerical Fluid Dynamics.
M. Brocchini D. H. Peregrine (1996) ArticleTitleIntegral flow properties of the swash zone and averaging J. Fluid Mech. 317 241–273
G. F. Carrier H. P. Greenspan (1958) ArticleTitleWater waves of finite amplitude on a sloping beach J. Fluid Mech. 4 97–109
B. Cockburn S.-Y. Lin C.-W. Shu (1989) ArticleTitleTVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems J. Comput. Phys. 84 90–113
B. Cockburn C.-W. Shu (1998) ArticleTitleThe Runge-Kutta discontinuous Galerkin finite element method for conservation laws V: Multidimensional systems J. Comput. Phys. 141 199–224 Occurrence Handle10.1006/jcph.1998.5892
J Jaffre C. Johnson A. Szepessy (1995) ArticleTitleConvergence of the discontinuous \mbox{Galerkin} finite element method for hyperbolic conservation laws Math. Models Methods Appl. Sci. 5 367–386 Occurrence Handle10.1142/S021820259500022X
L. Krivodonova J. Xin J.-F. Remacle N. Chevaugeon J.E. Flaherty (2004) ArticleTitleShock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws Appl. Numer. Math. 48 323–348 Occurrence Handle10.1016/j.apnum.2003.11.002 Occurrence HandleMR2056921
J. Pedlosky (1987) Geophysical Fluid Dynamics Springer New York 710
D. H. Peregrine S. M. Williams (2001) ArticleTitleSwash overtopping a truncated plane beach J. Fluid Mech. 440 391–399 Occurrence Handle10.1017/S002211200100492X
B. Perthame (1999) An introduction to kinetic schemes for gas dynamics D. Kröner M. Ohlberger C. Rohde (Eds) An Introduction to Recent Developments in Theory and Numerics for Conservation Laws Springer-Verlag Berlin 1–27
B. Perthame C. Simeoni (2001) ArticleTitleA kinetic scheme for the Saint-Venant scheme with a source term Calcolo 38 201–231 Occurrence Handle10.1007/s10092-001-8181-3
Schwanenberg, D. (2003). Die Runge-Kutta-Discontinuous-Galerkin-Methode zur Lösung konvektionsdominierter tiefengemittelter Flachwasserprobleme, Ph.D. Dissertation Technischen Hochschule Aachen, p. 133.
M. C. Shen R. E. Meyer (1963) ArticleTitleClimb of a bore on a beach J. Fluid Mech. 16 113–125
C-W. Shu S. Osher (1989) ArticleTitleEfficient implementation of essentially non-oscillatory shock-capturing schemes II J. Comput. Phy. 83 32–78 Occurrence Handle10.1016/0021-9991(89)90222-2
J. Smoller (1994) Shock Waves and Reaction-Diffusion Equations Springer-Verlag New York 632
E. F. Toro (1999) Shock Capturing Methods for Free-Surface Flows Wiley Toronto 309
E. F. Toro M. Spruce W. Speares (1994) ArticleTitleRestoration of the contact surface in the HLL-Riemann solver Shock Waves 4 25–34 Occurrence Handle10.1007/BF01414629
J.J. W. Vegt Particlevan der H. Ven Particlevan der (2002) ArticleTitleSpace-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows I. General formulation J. Comput. Phy. 182 546–585 Occurrence Handle10.1006/jcph.2002.7185
W. R. Young (1986) ArticleTitleElliptical vortices in shallow water J. Fluid Mech. 171 101–119
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Bokhove, O. Flooding and Drying in Discontinuous Galerkin Finite-Element Discretizations of Shallow-Water Equations. Part 1: One Dimension. J Sci Comput 22, 47–82 (2005). https://doi.org/10.1007/s10915-004-4136-6
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DOI: https://doi.org/10.1007/s10915-004-4136-6