Cited By
View all- Dumbser MBusto SElena Vázquez-Cendón MPeshkov I(2023)Preface for the special issue “Hyperbolic PDE in computational physicsApplied Mathematics and Computation10.1016/j.amc.2023.127994450:COnline publication date: 10-May-2023
A geometrically intrinsic lagrangian-Eulerian scheme for 2D shallow water equations with variable topography and discontinuous data
References
[1]
M. Nestler, I. Nitschke, A. Voigt, A finite element approach for vector- and tensor-valued surface PDEs, J. Comp. Phys. 389 (2019) 48–61.
[2]
E. Bachini, M. Putti, Geometrically intrinsic modeling of shallow water flows, Math. Model. Num. Anal. 54 (6) (2020) 2125–2157.
[3]
M.P. Neilson, J.A. Mackenzie, S.D. Webb, R.H. Insall, Modeling cell movement and chemotaxis using pseudopod-based feedback, SIAM J. Sci. Comput. 33 (3) (2011) 1035–1057.
[4]
J. Lowengrub, J. Allard, S. Aland, Numerical simulation of endocytosis: viscous flow driven by membranes with non-uniformly distributed curvature-inducing molecules, J. Comp. Phys. 309 (2016) 112–128.
[5]
I. Nitschke, A. Voigt, J. Wensch, A finite element approach to incompressible two-phase flow on manifolds, J. Fluid Mech. 708 (2012) 418.
[6]
F. Bouchut, M. Westdickenberg, Gravity driven shallow water models for arbitrary topography, Comm. Math. Sci. 2 (3) (2004) 359–389.
[7]
A. Decoene, L. Bonaventura, E. Miglio, F. Saleri, Asymptotic derivation of the section-averaged shallow water equations for natural river hydraulics, Math. Mod. Meth. Appl. Sci. 19 (03) (2009) 387–417.
[8]
F. Bouchut, S. Boyaval, A new model for shallow viscoelastic fluids, Math. Mod. Meth. Appl. Sci. 23 (8) (2013) 1479–1526.
[9]
I. Fent, M. Putti, C. Gregoretti, S. Lanzoni, Modeling shallow water flows on general terrains, Adv. Water Resour. 121 (2018) 316–332.
[10]
J.R. Holton, An introduction to dynamic meteorology, Burlington, MA: Elsevier Academic Press, 2004.
[11]
R.L. Higdon, Numerical modelling of ocean circulation, Acta Num. 15 (2006) 385.
[12]
C. Vreugdenhil, Numerical methods for shallow-water flow, Water Science and Technology Library, Springer Netherlands, 1994.
[13]
L. Bonaventura, E.D. Fernández-Nieto, J. Garres-Díaz, G. Narbona-Reina, Multilayer shallow water models with locally variable number of layers and semi-implicit time discretization, J. Comp. Phys. 364 (2018) 209–234.
[14]
R.L. Higdon, An automatically well-balanced formulation of pressure forcing for discontinuous Galerkin methods for the shallow water equations, J. Comp. Phys. (2022) 111102.
[15]
S. Lanzoni, A. Siviglia, A. Frascati, G. Seminara, Long waves in erodible channels and morphodynamic influence, Water Resour. Res. 42 (2006) W06D17.
[16]
R.M. Iverson, D.L. George, A depth-averaged debris-flow model that includes the effects of evolving dilatancy. i. physical basis, Proc. R. Soc. London 470 (2170) (2014) 20130819.
[17]
S. Noelle, N. Pankratz, G. Puppo, J.R. Natvig, Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows, J. Comp. Phys. 213 (2) (2006) 474–499.
[18]
M.J. Castro, A. Pardo Milanés, C. Parés, Well-balanced numerical schemes based on a generalized hydrostatic reconstruction technique, Math. Mod. Meth. Appl. Sci. 17 (12) (2007) 2055–2113.
[19]
J.M. Gallardo, C. Parés, M. Castro, On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas, J. Comp. Phys. 227 (1) (2007) 574–601.
[20]
M. Castro Díaz, J. López-García, C. Parés, High order exactly well-balanced numerical methods for shallow water systems, J. Comp. Phys. 246 (2013) 242–264.
[21]
J.P. Berberich, P. Chandrashekar, C. Klingenberg, High order well-balanced finite volume methods for multi-dimensional systems of hyperbolic balance laws, Comput. Fluids 219 (2021) 104858.
[22]
E. Godlewski, P.-A. Raviart, Numerical approximation of hyperbolic systems of conservation laws, volume 118 (Applied Mathematical Sciences Series), Springer, 2021 (Second Edition).
[23]
I. Gómez-Bueno, S. Boscarino, M.J. Castro, C. Parés, G. Russo, Implicit and semi-implicit well-balanced finite-volume methods for systems of balance laws, Appl. Num. Math. (Available online 30 September 2022).
[24]
S.B. Savage, K. Hutter, The dynamics of avalanches of granular materials from initiation to runout. part i: analysis, Acta Mech. 86 (1–4) (1991) 201–223.
[25]
J.A. Rossmanith, D.S. Bale, R.J. LeVeque, A wave propagation algorithm for hyperbolic systems on curved manifolds, J. Comp. Phys. 199 (2) (2004) 631–662.
[26]
D.S. Bale, R.J. Leveque, S. Mitran, J.A. Rossmanith, A wave propagation method for conservation laws and balance laws with spatially varying flux functions, SIAM J. Sci. Comput. 24 (3) (2003) 955–978.
[27]
B. Andreianov, K.H. Karlsen, N.H. Risebro, On vanishing viscosity approximation of conservation laws with discontinuous flux, NHM 5 (3) (2010) 617–633.
[28]
B. Andreianov, K.H. Karlsen, N.H. Risebro, A theory of l 1-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal. 201 (1) (2011) 27–86.
[29]
X. Sun, G. Wang, Y. Ma, A new modified local Lax-Friedrichs scheme for scalar conservation laws with discontinuous flux, Appl. Math. Let. 105 (2020) 106328.
[30]
D. Qiao, Z. Lin, M. Guo, X. Yang, X. Li, P. Zhang, X. Zhang, Riemann solvers of a conserved high-order traffic flow model with discontinuous fluxes, Appl. Math. Comput. 413 (2022) 126648.
[31]
F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-Balanced schemes for sources, Frontiers in Mathematics, volume 2/2004, Birkhäuser Basel, 2004.
[32]
M. Baldauf, Discontinuous Galerkin solver for the shallow-water equations in covariant form on the sphere and the ellipsoid, J. Comp. Phys. 410 (2020) 109384.
[33]
M.G. Carlino, E. Gaburro, Well balanced finite volume schemes for shallow water equations on manifolds, Appl. Math. Comput. 441 (2023) 127676.
[34]
E. Bachini, Numerical Methods for Shallow Water Equations on Regular Surfaces, Department of Mathematics ”Tulli Levi-Civita”, Doctoral Program in Mathematics, University of Padua, Italy, 2020, Ph.D. thesis.
[35]
D. Levy, G. Puppo, G. Russo, Central weno schemes for hyperbolic systems of conservation laws, ESAIM-Math. Model. Num. 33 (3) (1999) 547–571.
[36]
A. Kurganov, E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations, J. Comp. Phys. 160 (1) (2000) 241–282.
[37]
M. Dumbser, W. Boscheri, M. Semplice, G. Russo, Central weighted eno schemes for hyperbolic conservation laws on fixed and moving unstructured meshes, SIAM J. Sci. Comput. 39 (6) (2017) A2564–A2591.
[38]
E.F. Toro, A. Hidalgo, M. Dumbser, FORCE Schemes on unstructured meshes i: conservative hyperbolic systems, J. Comp. Phys. 228 (9) (2009) 3368–3389.
[39]
C.-S. Huang, T. Arbogast, C.-H. Hung, A semi-Lagrangian finite difference WENO scheme for scalar nonlinear conservation laws, J. Comp. Phys. 322 (2016) 559–585.
[40]
R. Loubere, P.-H. Maire, M. Shashkov, J. Breil, S. Galera, ReALE: a reconnection-based arbitrary-Lagrangian–Eulerian method, J. Comp. Phys. 229 (12) (2010) 4724–4761.
[41]
N.R. Morgan, M.A. Kenamond, D.E. Burton, T.C. Carney, D.J. Ingraham, An approach for treating contact surfaces in Lagrangian cell-centered hydrodynamics, J. Comp. Phys. 250 (2013) 527–554.
[42]
R. Loubere, M. Dumbser, S. Diot, A new family of high order unstructured MOOD and ADER finite volume schemes for multidimensional systems of hyperbolic conservation laws, Commun. Comput. Phys. 16 (3) (2014) 718–763.
[43]
W. Boscheri, R. Loubère, M. Dumbser, Direct arbitrary-Lagrangian-Eulerian ADER-MOOD finite volume schemes for multidimensional hyperbolic conservation laws, J. Comp. Phys. 292 (2015) 56–87.
[44]
V.A. Dobrev, T. Ellis, T.V. Kolev, R. Rieben, Curvilinear finite elements for Lagrangian hydrodynamics, Int. J. Numer. Methods Fluids 65 (11–12) (2011) 1295–1310.
[45]
V.A. Dobrev, T.V. Kolev, R.N. Rieben, High-order curvilinear finite element methods for Lagrangian hydrodynamics, SIAM J. Sci. Comput. 34 (5) (2012) B606–B641.
[46]
V.A. Dobrev, T.E. Ellis, T.V. Kolev, R.N. Rieben, High-order curvilinear finite elements for axisymmetric Lagrangian hydrodynamics, Comput. Fluids 83 (2013) 58–69.
[47]
N.R. Morgan, K.N. Lipnikov, D.E. Burton, M.A. Kenamond, A Lagrangian staggered grid Godunov-like approach for hydrodynamics, J. Comp. Phys. 259 (2014) 568–597.
[48]
E. Gaburro, M. Dumbser, M.J. Castro, Direct arbitrary-Lagrangian-Eulerian finite volume schemes on moving nonconforming unstructured meshes, Comput. Fluids 159 (2017) 254–275.
[49]
X. Liu, N.R. Morgan, D.E. Burton, A Lagrangian discontinuous Galerkin hydrodynamic method, Comput. Fluids 163 (2018) 68–85.
[50]
E. Gaburro, W. Boscheri, S. Chiocchetti, C. Klingenberg, V. Springel, M. Dumbser, High order direct arbitrary-Lagrangian-Eulerian schemes on moving voronoi meshes with topology changes, J. Comp. Phys. 407 (2020) 109167.
[51]
N.R. Morgan, B.J. Archer, On the origins of Lagrangian hydrodynamic methods, Nucl. Technol. 207 (sup1) (2021) S147–S175.
[52]
J. Douglas, F. Pereira, L.-M. Yeh, A locally conservative Eulerian-Lagrangian numerical method and its application to nonlinear transport in porous media, Comput. Geosci. 4 (1) (2000) 1–40.
[53]
J. Douglas, F. Pereira, L.-M. Yeh, A Locally Conservative Eulerian-Lagrangian Method for Flow in a Porous Medium of a Mixture of Two Components Having Different Densities, Numerical Treatment of Multiphase Flows in Porous Media, Springer, 2000, pp. 138–155.
[54]
J. Douglas, C.-S. Huang, A locally conservative Eulerian-Lagrangian finite difference method for a parabolic equation, BIT Numer. Math. 41 (3) (2001) 480–489.
[55]
J. Pérez, Lagrangian-Eulerian approximation methods for balance laws and hyperbolic conservation laws, University of Campinas, 2015, Ph.D. thesis.
[56]
E. Abreu, W. Lambert, J. Perez, A. Santo, A new finite volume approach for transport models and related applications with balancing source terms, Math. Comp. Simul. 137 (2017) 2–28.
[57]
E. Abreu, J. Pérez, A. Santo, Lagrangian–Eulerian approximation methods for balance laws and hyperbolic conservation laws, Rev. UIS Ing. 17 (1) (2018) 191–200.
[58]
E. Abreu, J. Pérez, A. Santo, A conservative Lagrangian–Eulerian finite volume approximation method for balance law problems, Proc. Ser. Braz. Soc. Comput. Appl. Math. 6 (1) (2018).
[59]
E. Abreu, J. François, W. Lambert, J. Pérez, A class of positive semi-discrete Lagrangian–Eulerian schemes for multidimensional systems of hyperbolic conservation laws, J. Scient. Comput. 90 (1) (2022) 1–79.
[60]
E. Abreu, J. François, W. Lambert, J. Pérez, A semi-discrete Lagrangian–Eulerian scheme for hyperbolic-transport models, J. Comput. Appl. Math. 406 (2022) 114011.
[61]
E. Abreu, J. Pérez, A fast, robust, and simple Lagrangian–Eulerian solver for balance laws and applications, Comput. Math. Appl. 77 (9) (2019) 2310–2336.
[62]
E. Abreu, W. Lambert, J. Pérez, A. Santo, A weak asymptotic solution analysis for a Lagrangian–Eulerian scheme for scalar hyperbolic conservation laws, Proceedings of the Seventeenth International Conference on Hyperbolic Problems, 2020, pp. 223–230.
[63]
E. Abreu, C. Diaz, J. Galvis, J. Pérez, On the conservation properties in multiple scale coupling and simulation for Darcy flow with hyperbolic-transport in complex flows, Multiscale Model. Simul. 18 (4) (2020) 1375–1408.
[64]
T. Barth, R. Herbin, M. Ohlberger, Finite Volume Methods: Foundation and Analysis, vol. 010329-1, John Wiley & Sons, Ltd, 2017, pp. 1–60.
[65]
R.J. Leveque, Finite volume methods for hyperbolic problems, Cambridge University Press, Trumpington Street, Cambridge, UK, 2002.
[66]
E. Abreu, V. Matos, J. Pérez, P. Rodríguez-Bermúdez, A class of Lagrangian–Eulerian shock-capturing schemes for first-order hyperbolic problems with forcing terms, J. Scient. Comput. 86 (1) (2021) 1–47.
[67]
E. Abreu, J. Perez, A. Santo, Solving hyperbolic conservation laws by using Lagrangian-Eulerian approach, Proc. Ser. Braz. Soc. Comput. Appl. Math. 5 ((1) 010329-1) (2017).
[68]
R. Eymard, T. Gallouët, R. Herbin, Convergence of a finite volume scheme for a nonlinear hyperbolic equation, Proceedings of the Third International Colloquium on Numerical Analysis, (Plovdiv, 1994)(Utrecht), VSP, 1995, pp. 61–70.
[69]
R. Eymard, T. Gallouët, R. Herbin, Existence and uniqueness of the entropy solution to a nonlinear hyperbolic equation, Chin. Ann. Math B16 (1995) 1–14.
[70]
C. Chainais-Hillairet, Finite volume schemes for a nonlinear hyperbolic equation. convergence towards the entropy solution and error estimate, Math. Model. Num. Anal. 33 (1) (1999) 129–156.
[71]
M.G. Crandall, A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34 (149) (1980) 1–21.
Index Terms
- A geometrically intrinsic lagrangian-Eulerian scheme for 2D shallow water equations with variable topography and discontinuous data
Index terms have been assigned to the content through auto-classification.
Recommendations
Positivity-Preserving Well-Balanced Arbitrary Lagrangian–Eulerian Discontinuous Galerkin Methods for the Shallow Water Equations
AbstractIn this paper, we develop well-balanced arbitrary Lagrangian–Eulerian discontinuous Galerkin (ALE-DG) methods for the shallow water equations, which preserve not only the still water equilibrium but also the moving water equilibrium. Based on the ...
A staggered semi-implicit spectral discontinuous Galerkin scheme for the shallow water equations
A spatially arbitrary high order, semi-implicit spectral discontinuous Galerkin (DG) scheme for the numerical solution of the shallow water equations on staggered control volumes is derived and discussed. The free surface elevation and the momentum are ...
A well-balanced van Leer-type numerical scheme for shallow water equations with variable topography
A well-balanced van Leer-type numerical scheme for the shallow water equations with variable topography is presented. The model involves a nonconservative term, which often makes standard schemes difficult to approximate solutions in certain regions. ...
Comments
Information & Contributors
Information
Published In
Elsevier Inc.
Publisher
Elsevier Science Inc.
United States
Publication History
Published: 15 April 2023
Author Tags
Qualifiers
- Research-article
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- View Citations1Total Citations
- 0Total Downloads
- Downloads (Last 12 months)0
- Downloads (Last 6 weeks)0
Reflects downloads up to 10 Nov 2024
Other Metrics
Citations
Cited By
View all- Dumbser MBusto SElena Vázquez-Cendón MPeshkov I(2023)Preface for the special issue “Hyperbolic PDE in computational physicsApplied Mathematics and Computation10.1016/j.amc.2023.127994450:COnline publication date: 10-May-2023
View Options
View options
Get Access
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign in