RESEARCH ARTICLE
Wetting and Drying Treatments With Mesh Adaptation for Shallow Water Equations Using a Runge–Kutta Discontinuous Galerkin Method
Camille Poussel,
Mehmet Ersoy, Frédéric Golay,
Corresponding Author
Camille Poussel
IMATH, Université de Toulon, La Garde, France
Correspondence:
Camille Poussel ([email protected])
Search for more papers by this authorCamille Poussel,
Mehmet Ersoy, Frédéric Golay,
Corresponding Author
Camille Poussel
IMATH, Université de Toulon, La Garde, France
Correspondence:
Camille Poussel ([email protected])
Search for more papers by this authorFunding: This work was supported by Région Provence-Alpes-Côte d'Azur, France (Emploi Jeunes Dctorants).
ABSTRACT
This work is devoted to the numerical simulation of Shallow Water Equations involving dry areas, a moving shoreline and in the context of mesh adaptation. The space and time discretization using the Runge–Kutta Discontinuous Galerkin approach is applied to nonlinear hyperbolic Shallow Water Equations. Problems with dry areas are challenging for such methods. To counter this issue, special treatment is applied around the shoreline. This work compares three treatments, one based on Slope Modification, one based on p-adaptation and the last one based on eXtended Finite Element methods and mesh adaptation.
Conflicts of Interest
The authors declare no conflicts of interest.
Open Research
Data Availability Statement
The data that support the findings of this study are available from the corresponding author upon reasonable request.
References
- 1J. B. Clément, F. Golay, M. Ersoy, and D. Sous, “An Adaptive Strategy for Discontinuous Galerkin Simulations of Richards' Equation: Application to Multi-Materials Dam Wetting,” Advances in Water Resources 151 (2021): 103897, https://doi.org/10.1016/j.advwatres.2021.103897.
- 2A. de Saint-Venant, Théorie Du Mouvement Non Permanent Des Eaux, Avec Application Aux Crues Des Rivières et à l'introduction Des Marées Dans Leur Lit (Paris: Comptes Rendus Hebdomadaires Des Séances de l'Académie Des SciencesGauthier-Villars, 1871).
- 3F. Marche, “Derivation of a New Two-Dimensional Viscous Shallow Water Model With Varying Topography, Bottom Friction and Capillary Effects,” European Journal of Mechanics - B/Fluids 26, no. 1 (2007): 49–63, https://doi.org/10.1016/j.euromechflu.2006.04.007.
- 4B. Cockburn, S. Y. Lin, and C. W. Shu, “TVB Runge-Kutta Local Projection Discontinuous Galerkin Finite Element Method for Conservation Laws III: One-Dimensional Systems,” Journal of Computational Physics 84, no. 1 (1989): 90–113, https://doi.org/10.1016/0021-9991(89)90183-6.
- 5S. Osher, “Convergence of Generalized MUSCL Schemes,” SIAM Journal on Numerical Analysis 22, no. 5 (1985): 947–961.
- 6L. Krivodonova, “Limiters for High-Order Discontinuous Galerkin Methods,” Journal of Computational Physics 226, no. 1 (2007): 879–896, https://doi.org/10.1016/j.jcp.2007.05.011.
- 7A. Ern, S. Piperno, and K. Djadel, “A Well-Balanced Runge–Kutta Discontinuous Galerkin Method for the Shallow-Water Equations With Flooding and Drying,” International Journal for Numerical Methods in Fluids 58, no. 1 (2007): 1–25, https://doi.org/10.1002/fld.1674.
10.1002/fld.1674 Google Scholar
- 8A. Duran and F. Marche, “Recent Advances on the Discontinuous Galerkin Method for Shallow Water Equations With Topography Source Terms,” Computers and Fluids 101 (2014): 88–104, https://doi.org/10.1016/j.compfluid.2014.05.031.
- 9Y. Xing, X. Zhang, and C. W. Shu, “Positivity-Preserving High Order Well-Balanced Discontinuous Galerkin Methods for the Shallow Water Equations,” Advances in Water Resources 33, no. 12 (2010): 1476–1493, https://doi.org/10.1016/j.advwatres.2010.08.005.
- 10G. Tumolo, L. Bonaventura, and M. Restelli, “A Semi-Implicit, Semi-Lagrangian, p-Adaptive Discontinuous Galerkin Method for the Shallow Water Equations,” Journal of Computational Physics 232, no. 1 (2013): 46–67, https://doi.org/10.1016/j.jcp.2012.06.006.
- 11C. Eskilsson, “An Hp-Adaptive Discontinuous Galerkin Method for Shallow Water Flows,” International Journal for Numerical Methods in Fluids 67, no. 11 (2011): 1605–1623, https://doi.org/10.1002/fld.2434.
- 12N. Moës, J. Dolbow, and T. Belytschko, “A Finite Element Method for Crack Growth Without Remeshing,” International Journal for Numerical Methods in Engineering 46, no. 1 (1999): 131–150, https://doi.org/10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J.
- 13B. L. Karihaloo and Q. Z. Xiao, “Modelling of Stationary and Growing Cracks in FE Framework Without Remeshing: A State-Of-The-Art Review,” Computers and Structures 81, no. 3 (2003): 119–129, https://doi.org/10.1016/S0045-7949(02)00431-5.
- 14J. F. Gerbeau, “Perthame aB. Derivation of Viscous Saint-Venant System for Laminar Shallow Water; Numerical Validation,” Discrete and Continuous Dynamical Systems – B 1, no. 1 (2001): 89–102, https://doi.org/10.3934/dcdsb.2001.1.89.
- 15T. Weiyan, Properties of the 2-D System of Shallow-Water Equations (2-D SSWE). 55 of Shallow Water Hydrodynamics (Amsterdam, The Netherlands: Elsevier, 1992).
- 16E. Godlewski and P. A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws (New York, NY: Springer, 1996).
10.1007/978-1-4612-0713-9 Google Scholar
- 17T. Coupez and E. Hachem, “Solution of High-Reynolds Incompressible Flow With Stabilized Finite Element and Adaptive Anisotropic Meshing,” Computer Methods in Applied Mechanics and Engineering 267 (2013): 65–85, https://doi.org/10.1016/j.cma.2013.08.004.
- 18T. Altazin, M. Ersoy, F. Golay, D. Sous, and L. Yushchenko, “Numerical Investigation of BB-AMR Scheme Using Entropy Production as Refinement Criterion,” International Journal of Computational Fluid Dynamics 30, no. 3 (2016): 256–271, https://doi.org/10.1080/10618562.2016.1194977.
- 19F. Golay, M. Ersoy, L. Yushchenko, and D. Sous, “Block-Based Adaptive Mesh Refinement Scheme Using Numerical Density of Entropy Production for Three-Dimensional Two-Fluid Flows,” International Journal of Computational Fluid Dynamics 29, no. 1 (2015): 67–81, https://doi.org/10.1080/10618562.2015.1012161.
- 20K. Pons, “Modélisation des tsunamis: propagation et impact” (PhD thesis, Université de Toulon, 2018).
- 21M. Ersoy, F. Golay, and L. Yushchenko, “Adaptive Multiscale Scheme Based on Numerical Density of Entropy Production for Conservation Laws,” Open Mathematics 11, no. 8 (2013): 1396–1398, https://doi.org/10.2478/s11533-013-0252-6.
10.2478/s11533-013-0252-6 Google Scholar
- 22D. A. D. Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods (Berlin Heidelberg: Springer, 2012).
10.1007/978-3-642-22980-0 Google Scholar
- 23S. K. Godunov and I. Bohachevsky, “Finite Difference Method for Numerical Computation of Discontinuous Solutions of the Equations of Fluid Dynamics,” Matematičeskij Sbornik 47 (1959): 271–306.
- 24T. Gallouët, J. M. Hérard, and N. Seguin, “Some Approximate Godunov Schemes to Compute Shallow-Water Equations With Topography,” Computers and Fluids 32, no. 4 (2003): 479–513, https://doi.org/10.1016/S0045-7930(02)00011-7.
- 25A. Harten, P. D. Lax, and B. V. Leer, “On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws,” SIAM Review 25, no. 1 (1983): 35–61.
- 26A. Duran, “Numerical Simulation of Depth-Averaged Flow Models: A Class of Finite Volume and Discontinuous Galerkin Approaches” (PhD thesis, Université Montpellier II, 2014).
- 27K. Dutt and L. Krivodonova, “A High-Order Moment Limiter for the Discontinuous Galerkin Method on Triangular Meshes,” Journal of Computational Physics 433 (2021): 110188, https://doi.org/10.1016/j.jcp.2021.110188.
- 28C. W. Shu and S. Osher, “Efficient Implementation of Essentially Non-Oscillatory Shock-Capturing Schemes,” Journal of Computational Physics 77, no. 2 (1988): 439–471, https://doi.org/10.1016/0021-9991(88)90177-5.
- 29C. W. Shu, “Total-Variation-Diminishing Time Discretizations,” SIAM Journal on Scientific and Statistical Computing 9, no. 6 (1988): 1073–1084, https://doi.org/10.1137/0909073.
- 30S. Gottlieb, C. W. Shu, and E. Tadmor, “Strong Stability-Preserving High-Order Time Discretization Methods,” SIAM Review 43, no. 1 (2001): 89–112, https://doi.org/10.1137/s003614450036757x.
- 31B. Cockburn and C. W. Shu, “Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems,” Journal of Scientific Computing 16 (2001): 173–261, https://doi.org/10.1023/A:1012873910884.
10.1023/A:1012873910884 Google Scholar
- 32B. Cockburn and C. W. Shu, “The Runge-Kutta Local Projection P1-Discontinuous-Galerkin Finite Element Method for Scalar Conservation Laws,” ESAIM. Mathematical Modelling and Numerical Analysis 25, no. 3 (1991): 337–361, https://doi.org/10.1051/m2an/1991250303371.
- 33N. Chalmers and L. Krivodonova, “A Robust CFL Condition for the Discontinuous Galerkin Method on Triangular Meshes,” Journal of Computational Physics 403 (2020): 109095, https://doi.org/10.1016/j.jcp.2019.109095.
- 34C. Nielsen and C. Apelt, “Parameters Affecting the Performance of Wetting and Drying in a Two-Dimensional Finite Element Long Wave Hydrodynamic Model,” Journal of Hydraulic Engineering 129, no. 8 (2003): 628–636, https://doi.org/10.1061/(ASCE)0733-9429(2003)129:8(628).
10.1061/(ASCE)0733-9429(2003)129:8(628) Google Scholar
- 35T. C. Gopalakrishnan, “A Moving Boundary Circulation Model for Regions With Large Tidal Flats,” International Journal for Numerical Methods in Engineering 28, no. 2 (1989): 245–260, https://doi.org/10.1002/nme.1620280202.
- 36A. Balzano, “Evaluation of Methods for Numerical Simulation of Wetting and Drying in Shallow Water Flow Models,” Coastal Engineering 34, no. 1 (1998): 83–107, https://doi.org/10.1016/S0378-3839(98)00015-5.
- 37B. Perthame and C. Simeoni, “A Kinetic Scheme for the Saint-Venant System With a Source Term,” Calcolo 38, no. 4 (2001): 201–231, https://doi.org/10.1007/s10092-001-8181-3.
- 38H. Tang and G. Warnecke, “A Runge–Kutta Discontinuous Galerkin Method for the Euler Equations,” Computers and Fluids 34, no. 3 (2005): 375–398, https://doi.org/10.1016/j.compfluid.2004.01.004.
- 39M. Bristeau and B. Perthame, Kinetic Schemes for Solving Saint-Venant Equations on Unstructured Grids (Singapore: World Scientific, 2001), 267–277.
10.1142/9789812810816_0013 Google Scholar
- 40P. D. Bates and J. M. Hervouet, “A New Method for Moving–Boundary Hydrodynamic Problems in Shallow Water,” Proceedings of the Royal Society of London, Series A: Mathematical, Physical and Engineering Sciences 455, no. 1988 (1999): 3107–3128, https://doi.org/10.1098/rspa.1999.0442.
- 41S. Bradford and B. Sanders, “Finite-Volume Model for Shallow-Water Flooding of Arbitrary Topography,” Journal of Hydraulic Engineering-ASCE 128, no. 3 (2002): 289–298, https://doi.org/10.1061/(ASCE)0733-9429(2002)128:3(289).
- 42N. Moës and T. Belytschko, X-FEM, de Nouvelles Frontières Pour Les Éléments Finis, vol. 11 (Giens, France: CSMA, 2001), 305–318.
- 43T. P. Fries and T. Belytschko, “The Extended/Generalized Finite Element Method: An Overview of the Method and Its Applications,” International Journal for Numerical Methods in Engineering 84, no. 3 (2010): 253–304, https://doi.org/10.1002/nme.2914.
- 44S. Bunya, E. J. Kubatko, J. J. Westerink, and C. Dawson, “A Wetting and Drying Treatment for the Runge–Kutta Discontinuous Galerkin Solution to the Shallow Water Equations,” Computer Methods in Applied Mechanics and Engineering 198, no. 17–20 (2009): 1548–1562, https://doi.org/10.1016/j.cma.2009.01.008.
- 45Y. Xing and X. Zhang, “Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for the Shallow Water Equations on Unstructured Triangular Meshes,” Journal of Scientific Computing 57, no. 1 (2013): 19–41, https://doi.org/10.1007/s10915-013-9695-y.
- 46X. Zhang, Y. Xia, and C. W. Shu, “Maximum-Principle-Satisfying and Positivity-Preserving High Order Discontinuous Galerkin Schemes for Conservation Laws on Triangular Meshes,” Journal of Scientific Computing 50, no. 1 (2012): 29–62, https://doi.org/10.1007/s10915-011-9472-8.
- 47S. Vater, N. Beisiegel, and J. Behrens, “A Limiter-Based Well-Balanced Discontinuous Galerkin Method for Shallow-Water Flows With Wetting and Drying: One-Dimensional Case,” Advances in Water Resources 85 (2015): 1–13, https://doi.org/10.1016/j.advwatres.2015.08.008.
- 48S. Vater, N. Beisiegel, and J. Behrens, “A Limiter-Based Well-Balanced Discontinuous Galerkin Method for Shallow-Water Flows With Wetting and Drying: Triangular Grids,” International Journal for Numerical Methods in Fluids 91, no. 8 (2019): 395–418, https://doi.org/10.1002/fld.4762.
- 49P. Mossier, A. Beck, and C. D. Munz, “A P-Adaptive Discontinuous Galerkin Method With Hp-Shock Capturing,” Journal of Scientific Computing 91, no. 1 (2022): 4, https://doi.org/10.1007/s10915-022-01770-6.
- 50L. Wang and D. J. Mavriplis, “Adjoint-Based h–p Adaptive Discontinuous Galerkin Methods for the 2D Compressible Euler Equations,” Journal of Computational Physics 228, no. 20 (2009): 7643–7661, https://doi.org/10.1016/j.jcp.2009.07.012.
- 51E. J. Kubatko, S. Bunya, C. Dawson, and J. J. Westerink, “Dynamic P-Adaptive Runge–Kutta Discontinuous Galerkin Methods for the Shallow Water Equations,” Computer Methods in Applied Mechanics and Engineering 198, no. 21 (2009): 1766–1774, https://doi.org/10.1016/j.cma.2009.01.007.
- 52T. Belytschko and T. Black, “Elastic Crack Growth in Finite Elements With Minimal Remeshing,” International Journal for Numerical Methods in Engineering 45, no. 5 (1999): 601–620, https://doi.org/10.1002/(SICI)1097-0207(19990620)45:5<601::AID-NME598>3.0.CO;2-S.
- 53G. F. Carrier and H. P. Greenspan, “Water Waves of Finite Amplitude on a Sloping Beach,” Journal of Fluid Mechanics 4, no. 1 (1958): 97–109, https://doi.org/10.1017/S0022112058000331.
- 54O. Bokhove, “Flooding and Drying in Discontinuous Galerkin Finite-Element Discretizations of Shallow-Water Equations. Part 1: One Dimension,” Journal of Scientific Computing 22, no. 1 (2005): 47–82, https://doi.org/10.1007/s10915-004-4136-6.
- 55V. Roeber, K. F. Cheung, and M. H. Kobayashi, “Shock-Capturing Boussinesq-Type Model for Nearshore Wave Processes,” Coastal Engineering 57, no. 4 (2010): 407–423, https://doi.org/10.1016/j.coastaleng.2009.11.007.
- 56V. Roeber and K. F. Cheung, “Boussinesq-Type Model for Energetic Breaking Waves in Fringing Reef Environment,” Coastal Engineering 70 (2012): 1–20, https://doi.org/10.1016/j.coastaleng.2012.06.001.
- 57M. Kazolea, A. I. Delis, and C. E. Synolakis, “Numerical Treatment of Wave Breaking on Unstructured Finite Volume Approximations for Extended Boussinesq-Type Equations,” Journal of Computational Physics 271 (2014): 281–305, https://doi.org/10.1016/j.jcp.2014.01.030.
- 58B. Bonev, J. S. Hesthaven, F. X. Giraldo, and M. A. Kopera, “Discontinuous Galerkin Scheme for the Spherical Shallow Water Equations With Applications to Tsunami Modeling and Prediction,” Journal of Computational Physics 362 (2018): 425–448, https://doi.org/10.1016/j.jcp.2018.02.008.
- 59M. Dumbser and V. Casulli, “A Staggered Semi-Implicit Spectral Discontinuous Galerkin Scheme for the Shallow Water Equations,” Applied Mathematics and Computation 219, no. 15 (2013): 8057–8077, https://doi.org/10.1016/j.amc.2013.02.041.
- 60P. J. Lynett, D. Swigler, S. Son, D. Bryant, and S. Socolofsky, “Experimental Study of Solitary Wave Evolution Over a 3D Shallow Shelf,” Coastal Engineering Proceedings 32 (2010): 1, https://doi.org/10.9753/icce.v32.currents.1.
10.9753/icce.v32.currents.1 Google Scholar
- 61F. Shi, J. T. Kirby, J. C. Harris, J. D. Geiman, and S. T. Grilli, “A High-Order Adaptive Time-Stepping TVD Solver for Boussinesq Modeling of Breaking Waves and Coastal Inundation,” Ocean Modelling 43–44 (2012): 36–51, https://doi.org/10.1016/j.ocemod.2011.12.004.
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