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A high-order explicit Runge-Kutta approximation technique for the Shallow Water Equations

Published 25 Mar 2024 in math.NA and cs.NA | (2403.17123v1)

Abstract: We introduce a high-order space-time approximation of the Shallow Water Equations with sources that is invariant-domain preserving (IDP) and well-balanced with respect to rest states. The employed time-stepping technique is a novel explicit Runge-Kutta (ERK) approach which is an extension of the class of ERK-IDP methods introduced by Ern and Guermond (SIAM J. Sci. Comput. 44(5), A3366--A3392, 2022) for systems of non-linear conservation equations. The resulting method is then numerically illustrated through verification and validation.

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References (51)
  1. The deal.II library, version 9.5. Journal of Numerical Mathematics, 31(3):231–246, 2023.
  2. E. Audusse and M.-O. Bristeau. A well-balanced positivity preserving “second-order” scheme for shallow water flows on unstructured meshes. J. Comput. Phys., 206(1):311–333, 2005.
  3. A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput., 25(6):2050–2065, 2004.
  4. Well-balanced second-order approximation of the shallow water equation with continuous finite elements. SIAM J. Numer. Anal., 55(6):3203–3224, 2017.
  5. A. Bermúdez and M. E. Vázquez. Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids, 23(8):1049–1071, 1994.
  6. Finite volume evolution Galerkin methods for the shallow water equations with dry beds. Commun. Comput. Phys., 10(2):371–404, 2011.
  7. Flux-corrected transport. Journal of computational physics, 135(2):172–186, 1997.
  8. F. Bouchut. Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2004.
  9. M.-O. Bristeau and B. Coussin. Boundary conditions for the shallow water equations solved by kinetic schemes. Research Report inria-00072305, INRIA, 2001.
  10. Efficient shallow water simulations on gpus: Implementation, visualization, verification, and validation. Computers & Fluids, 55:1–12, 2012.
  11. M. J. Castro and M. Semplice. Third-and fourth-order well-balanced schemes for the shallow water equations based on the cweno reconstruction. International Journal for Numerical Methods in Fluids, 89(8):304–325, 2019.
  12. SERGHEI (SERGHEI-SWE) v1.0: a performance-portable high-performance parallel-computing shallow-water solver for hydrology and environmental hydraulics. Geoscientific Model Development, 16(3):977–1008, 2023.
  13. Well-balanced positivity preserving central-upwind scheme for the shallow water system with friction terms. Internat. J. Numer. Methods Fluids, 78(6):355–383, 2015.
  14. Well-balanced schemes for the shallow water equations with coriolis forces. Numerische Mathematik, 138:939–973, 2018.
  15. Swashes: a compilation of shallow water analytic solutions for hydraulic and environmental studies. International Journal for Numerical Methods in Fluids, 72(3):269–300, 2013.
  16. V. Delmas and A. Soulaïmani. Multi-gpu implementation of a time-explicit finite volume solver using cuda and a cuda-aware version of openmpi with application to shallow water flows. Computer Physics Communications, 271:108190, 2022.
  17. Modeling hurricane waves and storm surge using integrally-coupled, scalable computations. Coastal Engineering, 58(1):45–65, 2011.
  18. Asymptotic preserving scheme for the shallow water equations with source terms on unstructured meshes. J. Comput. Phys., 287:184–206, 2015.
  19. A. Ern and J.-L. Guermond. Invariant-domain-preserving high-order time stepping: I. explicit runge–kutta schemes. SIAM Journal on Scientific Computing, 44(5):A3366–A3392, 2022.
  20. On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas. Journal of Computational Physics, 227(1):574 – 601, 2007.
  21. Strong stability-preserving high-order time discretization methods. SIAM Review, 43(1):89–112, 2001.
  22. On the theory of water waves. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 338(1612):43–55, 1974.
  23. A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal., 33(1):1–16, 1996.
  24. J.-L. Guermond and R. Pasquetti. A correction technique for the dispersive effects of mass lumping for transport problems. Computer Methods in Applied Mechanics and Engineering, 253:186 – 198, 2013.
  25. J.-L. Guermond and B. Popov. Invariant domains and first-order continuous finite element approximation for hyperbolic systems. SIAM J. Numer. Anal., 54(4):2466–2489, 2016.
  26. A second-order maximum principle preserving Lagrange finite element technique for nonlinear scalar conservation equations. SIAM J. Numer. Anal., 52(4):2163–2182, 2014.
  27. Well-balanced second-order finite element approximation of the shallow water equations with friction. SIAM Journal on Scientific Computing, 40(6):A3873–A3901, 2018.
  28. Invariant domain preserving discretization-independent schemes and convex limiting for hyperbolic systems. Computer Methods in Applied Mechanics and Engineering, 347:143–175, 2019a.
  29. Robust explicit relaxation technique for solving the Green-Naghdi equations. J. Comput. Phys., 399:108917, 17, 2019b.
  30. On the implementation of a robust and efficient finite element-based parallel solver for the compressible navier-stokes equations. Computer Methods in Applied Mechanics and Engineering, 389:114250, 2022.
  31. H. Hajduk and D. Kuzmin. Bound-preserving and entropy-stable algebraic flux correction schemes for the shallow water equations with topography. arXiv preprint arXiv:2207.07261, 2022.
  32. On upstream differencing and godunov-type schemes for hyperbolic conservation laws. SIAM review, 25(1):35–61, 1983.
  33. M. Kawahara and T. Umetsu. Finite element method for moving boundary problems in river flow. International Journal for Numerical Methods in Fluids, 6(6):365–386, 1986.
  34. A. Kurganov and G. Petrova. A second-order well-balanced positivity preserving central-upwind scheme for the Saint-Venant system. Commun. Math. Sci., 5(1):133–160, 2007.
  35. Flux-corrected transport: principles, algorithms, and applications. Springer, 2012.
  36. Bound-preserving flux limiting for high-order explicit runge–kutta time discretizations of hyperbolic conservation laws. Journal of Scientific Computing, 91(1):21, 2022.
  37. Q. Liang and F. Marche. Numerical resolution of well-balanced shallow water equations with complex source terms. Advances in water resources, 32(6):873–884, 2009.
  38. M. Maier and M. Kronbichler. Efficient parallel 3d computation of the compressible euler equations with an invariant-domain preserving second-order finite-element scheme. ACM Transactions on Parallel Computing, 8(3):16:1–30, 2021.
  39. Towards transient experimental water surfaces: A new benchmark dataset for 2D shallow water solvers. Advances in Water Resources, 121:130–149, 2018.
  40. H. Nessyahu and E. Tadmor. Non-oscillatory central differencing for hyperbolic conservation laws. Journal of computational physics, 87(2):408–463, 1990.
  41. High-order well-balanced finite volume WENO schemes for shallow water equation with moving water. J. Comput. Phys., 226(1):29–58, 2007.
  42. B. Perthame and C. Simeoni. A kinetic scheme for the Saint-Venant system with a source term. Calcolo, 38(4):201–231, 2001.
  43. M. Ricchiuto and A. Bollermann. Stabilized residual distribution for shallow water simulations. J. Comput. Phys., 228(4):1071–1115, 2009.
  44. Application of conservative residual distribution schemes to the solution of the shallow water equations on unstructured meshes. Journal of Computational Physics, 222(1):287–331, 2007.
  45. Two barriers on strong-stability-preserving time discretization methods. Journal of Scientific Computing, 17:211–220, 2002.
  46. F. Serre. Contribution à l’étude des écoulements permanents et variables dans les canaux. La Houille Blanche, 39(6):830–872, 1953.
  47. C.-W. Shu and S. Osher. Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys., 77(2):439 – 471, 1988.
  48. W. C. Thacker. Some exact solutions to the nonlinear shallow-water wave equations. Journal of Fluid Mechanics, 107:499–508, 1981.
  49. United States Geological Survey. United States Geological Survey 3D Elevation Program: 1 Meter Digital Elevation Model, 2021. Distributed by OpenTopography.
  50. Y. Xing and C.-W. Shu. A survey of high order schemes for the shallow water equations. J. Math. Study, 47(3):221–249, 2014.
  51. S. T. Zalesak. Fully multidimensional flux-corrected transport algorithms for fluids. Journal of computational physics, 31(3):335–362, 1979.

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