Open Access
Issue
ESAIM: M2AN
Volume 58, Number 2, March-April 2024
Page(s) 723 - 757
DOI https://doi.org/10.1051/m2an/2024012
Published online 19 April 2024
  1. A.-J.-C. Barré 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. Comptes rendus hebdomadaires des séances de l’Académie des sciences (1871). [Google Scholar]
  2. S. Benzoni-Gavage and D. Serre, Multi-dimensional Hyperbolic Partial Differential Equations: First-order Systems and Applications. Oxford University Press on Demand (2007). [Google Scholar]
  3. A. Bermudez and M.E. Vazquez, Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23 (1994) 1049–1071. [Google Scholar]
  4. S. Bryson, Y. Epshteyn, A. Kurganov and G. Petrova, Well-balanced positivity preserving central-upwind scheme on triangular grids for the Saint-Venant system. ESAIM: Math. Modell. Numer. Anal. 45 (2011) 423–446. [Google Scholar]
  5. C. Chen, C. Dawson and E. Valseth, Cross-mode stabilized stochastic shallow water systems using stochastic finite element methods. Comput. Methods Appl. Mech. Eng. 405 (2023) 115873. [Google Scholar]
  6. A. Chertock, S. Cui, A. Kurganov and T. Wu, Well-balanced positivity preserving central-upwind scheme for the shallow water system with friction terms. Int. J. Numer. Methods Fluids 78 (2015) 355–383. [Google Scholar]
  7. A. Chertock, S. Jin and A. Kurganov, A well-balanced operator splitting based stochastic Galerkin method for the one-dimensional Saint-Venant system with uncertainty. Preprint https://chertock.wordpress.ncsu.edu/files/2019/10/CJK2.pdf (2015). [Google Scholar]
  8. A. Cohen and R. Moussa, Approximation of high-dimensional parametric PDEs. Acta Numer. 24 (2015) 1–159. [Google Scholar]
  9. N. Črnjarić-Žic, S. Vuković and L. Sopta, Balanced finite volume WENO and central WENO schemes for the shallow water and the open-channel flow equations. J. Comput. Phys. 200 (2004) 512–548. [Google Scholar]
  10. C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics. Springer (2016). [Google Scholar]
  11. D. Dai, Y. Epshteyn and A. Narayan, Hyperbolicity-preserving and well-balanced stochastic Galerkin method for shallow water equations. SIAM J. Sci. Comput. 43 (2021) A929–A952. [Google Scholar]
  12. D. Dai, Y. Epshteyn and A. Narayan, Hyperbolicity-preserving and well-balanced stochastic Galerkin method for two-dimensional shallow water equations. J. Comput. Phys. 452 (2022) 110901. [Google Scholar]
  13. B.J. Debusschere, H.N. Najm, P.P. Pébay, O.M. Knio, R.G. Ghanem and O.P. Le Maître, Numerical challenges in the use of polynomial chaos representations for stochastic processes. SIAM J. Sci. Comput. 26 (2004) 698–719. [Google Scholar]
  14. Y. Epshteyn and T. Moussa, Adaptive central-upwind scheme on triangular grids for the Saint-Venant system. Commun. Math. Sci. 21 (2023) 671–708. [Google Scholar]
  15. O.G. Ernst, A. Mugler, H.-J. Starkloff and E. Ullmann, On the convergence of generalized polynomial chaos expansions. ESAIM: Math. Modell. Numer. Anal. 46 (2012) 317–339. [Google Scholar]
  16. U. Fjordholm, S. Mishra and E. Tadmor, Energy Preserving and Energy Stable Schemes for the Shallow Water Equations. London Mathematical Society Lecture Note Series. Cambridge University Press (2009) 93–139. [Google Scholar]
  17. U.S. Fjordholm, S. Mishra and E. Tadmor, Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography. J. Comput. Phys. 230 (2011) 5587–5609. [Google Scholar]
  18. U.S. Fjordholm, S. Mishra and E. Tadmor, Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws. SIAM J. Numer. Anal. 50 (2012) 544–573. [Google Scholar]
  19. S. Gerster and M. Moussa, Entropies and symmetrization of hyperbolic stochastic Galerkin formulations. Commun. Comput. Phys. 27 (2020) 639–671. [Google Scholar]
  20. S. Gerster, A. Sikstel and G. Visconti, Haar-type stochastic Galerkin formulations for hyperbolic systems with Lipschitz continuous flux function. Preprint arXiv:2203.11718 (2022). [Google Scholar]
  21. S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods. SIAM Rev. 43 (2001) 89–112. [Google Scholar]
  22. S. Jin, D. Xiu and X. Zhu, A well-balanced stochastic galerkin method for scalar hyperbolic balance laws with random inputs. J. Sci. Comput. 67 (2016) 1198–1218. [Google Scholar]
  23. A. Kurganov, Finite-volume schemes for shallow-water equations. Acta Numer. 27 (2018) 289–351. [Google Scholar]
  24. A. Kurganov and D. Moussa, Central-upwind schemes for the Saint-Venant system. ESAIM: Math. Modell. Numer. Anal. 36 (2002) 397–425. [Google Scholar]
  25. A. Kurganov and G. Moussa, A second-order well-balanced positivity preserving central-upwind scheme for the Saint-Venant system. Commun. Math. Sci. 5 (2007) 133–160. [Google Scholar]
  26. J. Kusch, R.G. McClarren and M. Frank, Filtered stochastic Galerkin methods for hyperbolic equations. J. Comput. Phys. 403 (2020) 109073. [Google Scholar]
  27. R.J. LeVeque, Numerical Methods for Conservation Laws. Springer (1992). [Google Scholar]
  28. R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press (2002). [Google Scholar]
  29. X. Liu, J. Albright, Y. Epshteyn and A. Kurganov, Well-balanced positivity preserving central-upwind scheme with a novel wet/dry reconstruction on triangular grids for the Saint-Venant system. J. Comput. Phys. 374 (2018) 213–236. [Google Scholar]
  30. G. Poëtte, B. Després, and D. Lucor, Uncertainty quantification for systems of conservation laws. J. Comput. Phys. 228 (2009) 2443–2467. [Google Scholar]
  31. R. Pulch and D. Moussa, Generalised polynomial chaos for a class of linear conservation laws. J. Sci. Comput. 51 (2012) 293–312. [Google Scholar]
  32. B.D. Rogers, A.G.L. Borthwick and P.H. Taylor, Mathematical balancing of flux gradient and source terms prior to using Roe’s approximate Riemann solver. J. Comput. Phys. 192 (2003) 422–451. [Google Scholar]
  33. L. Schlachter and F. Moussa, A hyperbolicity-preserving stochastic Galerkin approximation for uncertain hyperbolic systems of equations. J. Comput. Phys. 375 (2018) 80–98. [Google Scholar]
  34. R.C. Smith, Uncertainty Quantification: Theory, Implementation, and Applications. SIAM-Society for Industrial and Applied Mathematics, Philadelphia (2013). [Google Scholar]
  35. T.J. Sullivan. Introduction to Uncertainty Quantification. Vol. 63. Springer (2015). [Google Scholar]
  36. E. Tadmor, The numerical viscosity of entropy stable schemes for systems of conservation laws. I. Math. Comput. 49 (1987) 91–103. [Google Scholar]
  37. E. Tadmor, Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer. 12 (2003) 451–512. [Google Scholar]
  38. K. Wu, H. Tang and D. Xiu, A stochastic Galerkin method for first-order quasilinear hyperbolic systems with uncertainty. J. Comput. Phys. 345 (2017) 224–244. [Google Scholar]
  39. Y. Xing, High order finite volume WENO schemes for the shallow water flows through channels with irregular geometry. J. Comput. Appl. Math. 299 (2016) 229–244. [Google Scholar]
  40. Y. Xing, Chapter 13 – numerical methods for the nonlinear shallow water equations, in Handbook of Numerical Analysis. Vol. 18 of Handbook of Numerical Methods for Hyperbolic Problems, edited by R. Abgrall and C.-W. Shu. (2017) 361–384. [Google Scholar]
  41. Y. Xing and C.-W. Shu, High order finite difference WENO schemes with the exact conservation property for the shallow water equations. J. Comput. Phys. 208 (2005) 206–227. [Google Scholar]
  42. Y. Xing and C.-W. Shu, High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms. J. Comput. Phys. 214 (2006) 567–598. [Google Scholar]
  43. Y. Xing and C.-W. Shu, A new approach of high order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms. Commun. Comput. Phys. 1 (2006) 100–134. [Google Scholar]
  44. D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press (2010). [Google Scholar]
  45. X. Zhong and C.-W. Shu, Entropy stable Galerkin methods with suitable quadrature rules for hyperbolic systems with random inputs. J. Sci. Comput. 92 (2022) 14. [Google Scholar]
  46. T. Zhou and T. Moussa, Galerkin methods for stochastic hyperbolic problems using bi-orthogonal polynomials. J. Sci. Comput. 51 (2012) 274–292. [Google Scholar]
  47. J.G. Zhou, D.M. Causon, C.G. Mingham and D.M. Ingram, The surface gradient method for the treatment of source terms in the shallow-water equations. J. Comput. Phys. 168 (2001) 1–25. [Google Scholar]

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