The article describes the application of an original quasi-acoustic scheme for numerical solution of two-dimensional shallow-water equations with an uneven bottom. The numerical scheme partitions the linear reconstruction of the solution into small-perturbation blocks. The main advantage of the quasi-acoustic scheme is that it constructs the solution without using limiters, artificial regularizers, or any tuning parameters. The scheme is verified on a number of test and prototype problems.
Similar content being viewed by others
References
P. Glaister, “Approximate Riemann solutions of the shallow water equations,” ASCE J. Hydraulic Eng.,26, No. 3, 293–306 (1988).
R. Salmon, “Numerical solution of the two-layer shallow water equations with bottom topography,” J. Marine Research,60, 605–638 (2002).
P.-W. Li and C.-M. Fan, “Generalized finite difference method for two-dimensional shallow water equations,” Eng. Anal. with Boundary Elements, 80, 58–71 (2017).
H. Altai and P. Dreyfuss, “Numerical solutions for 2D depth-averaged shallow water equations,” Intern. Math. Forum,13, No. 2, 79–90 (2018).
M. F. Ahmad and M. S. Sulaiman, “Deficiency of finite difference methods for capturing shock waves and wave propagation over uneven bottom seabed,” Intern. J. Eng. & Technology,7, No. 3.28, 97–101 (2018).
I. M. Navon, “Finite-element simulation of the shallow water equations model on a limited-area domain,” Appl. Math. Modelling,3, 337–348 (1979).
M. Kawahara and T. Umetsu, “Finite element method for moving boundary problems in river flow,” Intern. J. Numer. Methods in Fluids,6, 365–386 (1986).
E. Harnet, D. Y. Le Roux, V. Legat, and F. Deleersnijder, “An efficient Eulerian finite element method for the shallow water equations,” Ocean Modelling,10, 115–136 (2005).
T. Zhang, F. Fang, C. C. Pain, C. Maksimovic, P. Feng, I. M. Navon, and P. D. Bates, “Application of a three-dimensional unstructured-mesh finite-element flooding model and comparison with two-dimensional approaches,” Water Resources Management,30, No. 7, 823–841 (2016).
P. Azerad, J.-L. Guermond, and B. Popov, “Well-balanced second-order approximation of the shallow water equation with continuous finite elements,” SIAM J. Numer. Analysis,55, No. 6, 3203–3224 (2017).
J.-L. Guermond, M. Q. De Luna, B. Popov, C. E. Kees, and M. W. Farthing, “Well-balanced second-order finite element approximation of the shallow water equations with friction,” SIAM J. Scient. Computing,40, No. 6, 3873–3901 (2018).
A. Bermudez and M. E. Vazquez, “Upwind methods for hyperbolic conservation laws with source terms,” Computers & Fluids,23, No. 8, 1049-1071 (1994).
P. Garcia-Navarro and M. E. Vazquez, “On numerical treatment of the source terms in shallow water equations,” Computers & Fluids,38, 221–234 (2009).
E. F. Toro, Shock-Capturing Methods for Free–Surface Shallow Flows, Wiliey, New York (2001).
Q. Liang and A. G. I. Borthwick, “Adaptive quadtree simulation of shallow flows with wet–dry fronts over complex topography,” Computers & Fluids,38, 221–234 (2009).
H. Yuxin, Z. Ningchuan, and P. Yuguo, “Well-balanced finite volume scheme for shallow water flooding and drying over arbitrary topography,” Eng. Appl. Comput. Fluid Mechanics,7, No. 1, 40–54 (2013).
B. Turan and K.-H. Wang, “An object-oriented overland flow solver for watershed flood inundation predictions: case study of Ulus basin, Turkey,” J. Hydrol. Hydromech.,62, No. 3, 209–217 (2014).
С. Berthon and С. Chalons, “A fully well-balanced, positive and entropy-satisfying Godunov-type method for the shallow-water equations,” Math. Comput.,85, No. 299, 1281–1307 (2016).
V. Michel-Dansac, C. Berthon, S. Clain, and F. Foucher, “A well-balanced scheme for the shallow-water equations with topography or Manning friction,” J. Comput. Physics,335, 115–154 (2017).
A. Kurganov, “Finite-volume schemes for shallow-water equations,” Acta Numerica, 289–351 (2018).
A. Harten, P. D. Lax, and B. Van Leer, “On upstream differencing and Godunov-type schemes for hyperbolic conservation laws,” SIAM Review,25, No. 1, 35–61 (1983).
E. F. Tor, Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, Berlin (1999).
S. K. Godunov, “Difference method for numerical calculation of discontinuous solutions of hydrodynamic equations,” Mat. Sborn.,47, No. 89, 3, 271–306 (1959).
M. V. Abakumov, A. M. Galanina, V. A. Isakov, N. N. Tyurina, A. P. Favorskii, and A. B. Khrulenko, “Quasi-acoustic scheme for Euler equations in hydrodynamics,” Diff. Uravn.,47, No. 7, 1092-1098 (2011).
V. A. Isakov and A. P. Favorskii, “Qusi-acoustic scheme for Euler equations of hydrodynamics in the case of two spatial dimensions,” Mat. Modelirovanie,24, No. 12, 55-59 (2012).
V. A. Isakov, “Application of a quasi-acoustic scheme to many-dimensional gas-dynamic problems,” Comput. Math. and Modeling,25, No. 3, 334–341 (2014).
V. A. Isakov, “Application of a quasi-acoustic scheme to solve shallow-water equations with an uneven bottom,” Comput. Math. and Modeling,29, No. 3, 319–333 (2018).
A. A. Samarskii, Theory of Finite-Difference Schemes [in Russian], Nauka, Moscow (1977).
A. S. Petrosyan, Additional Chapters in Shallow-Water Theory [in Russian], Facsimile, IKI RAN, Moscow (2014).
O. V. Bulatov, “Analytical solutions and numerical solutions of the Saint-Venant equations for some problems of the decay of a discontinuity above a cliff and a step on the bottom,” Zh. Vych. Mat. i Mat. Fiziki,54, No. 1, 150–165 (2014).
R. Bernetti, V. A. Titarev, and E. F. Toro, “Exact solution of the Riemann problem for the shallow water equations with discontinuous bottom geometry,” J. Comput. Physics,227, No. 6, 3212–3243 (2008).
R. J. LeVeque, “Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm,” J. Comput. Physics,146, No. 1, 346–365 (1998).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Prikladnaya Matematika i Informatika, No. 62, 2019, pp. 34–54.
Rights and permissions
About this article
Cite this article
Isakov, V.A. Application of a Limiterless Quasi Acoustic Scheme to Solve Two-Dimensional Shallow Water Equations with an Uneven Bottom. Comput Math Model 31, 25–42 (2020). https://doi.org/10.1007/s10598-020-09474-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10598-020-09474-y