A high-order explicit Runge-Kutta approximation technique for the shallow water equations

Guermond, Jean-Luc; Maier, Matthias; Tovar, Eric Joseph

doi:10.1016/j.compfluid.2024.106493

Title: A high-order explicit Runge-Kutta approximation technique for the shallow water equations

Journal Article · 26 November 2024 · Computers and Fluids

DOI: https://doi.org/10.1016/j.compfluid.2024.106493 · OSTI ID:2479480

Guermond, Jean-Luc ^[1];

^[1];

^[2]

Texas A & M Univ., College Station, TX (United States)
Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Here, we introduce a high-order space–time approximation of the Shallow Water Equations with sources that is invariant-domain preserving (IDP), well-balanced with respect to rest states, and employs a novel explicit Runge–Kutta (ERK) introduced in 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.

View Accepted Manuscript (DOE)

Cite

Export

Save

Research Organization:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: USDOE Laboratory Directed Research and Development (LDRD) Program

Grant/Contract Number:: 89233218CNA000001

OSTI ID:: 2479480

Report Number(s):: LA-UR--24-22352

Journal Information:: Computers and Fluids, Journal Name: Computers and Fluids Vol. 288; ISSN 0045-7930

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (50)

Finite element method for moving boundary problems in river flow Kawahara, Mutsuto; Umetsu, Tsuyoshi International Journal for Numerical Methods in Fluids, Vol. 6, Issue 6 https://doi.org/10.1002/fld.1650060605	journal	June 1986
SWASHES: a compilation of shallow water analytic solutions for hydraulic and environmental studies: ANALYTIC SOLUTIONS FOR SHALLOW WATER EQUATIONS Delestre, Olivier; Lucas, Carine; Ksinant, Pierre-Antoine International Journal for Numerical Methods in Fluids, Vol. 72, Issue 3 https://doi.org/10.1002/fld.3741	journal	October 2012
Well-balanced positivity preserving central-upwind scheme for the shallow water system with friction terms: Well-balanced positivity preserving central-upwind scheme for the shallow water system with friction terms Chertock, A.; Cui, S.; Kurganov, A. International Journal for Numerical Methods in Fluids, Vol. 78, Issue 6 https://doi.org/10.1002/fld.4023	journal	April 2015
Third‐ and fourth‐order well‐balanced schemes for the shallow water equations based on the CWENO reconstruction Castro, Manuel J.; Semplice, Matteo International Journal for Numerical Methods in Fluids, Vol. 89, Issue 8 https://doi.org/10.1002/fld.4700	journal	December 2018
Non‐oscillatory relaxation methods for the shallow‐water equations in one and two space dimensions Seaïd, Mohammed International Journal for Numerical Methods in Fluids, Vol. 46, Issue 5 https://doi.org/10.1002/fld.766	journal	August 2004
Flux-Corrected Transport Boris, Jay P.; Book, David L. Journal of Computational Physics, Vol. 135, Issue 2 https://doi.org/10.1006/jcph.1997.5700	journal	August 1997
Balancing Source Terms and Flux Gradients in High-Resolution Godunov Methods: The Quasi-Steady Wave-Propagation Algorithm LeVeque, Randall J. Journal of Computational Physics, Vol. 146, Issue 1 https://doi.org/10.1006/jcph.1998.6058	journal	October 1998
Well-balanced schemes for the shallow water equations with Coriolis forces Chertock, Alina; Dudzinski, Michael; Kurganov, Alexander Numerische Mathematik, Vol. 138, Issue 4 https://doi.org/10.1007/s00211-017-0928-0	journal	December 2017
A kinetic scheme for the Saint-Venant system¶with a source term Perthame, B.; Simeoni, C. Calcolo, Vol. 38, Issue 4 https://doi.org/10.1007/s10092-001-8181-3	journal	December 2001
Bound-preserving Flux Limiting for High-Order Explicit Runge–Kutta Time Discretizations of Hyperbolic Conservation Laws Kuzmin, Dmitri; Quezada de Luna, Manuel; Ketcheson, David I. Journal of Scientific Computing, Vol. 91, Issue 1 https://doi.org/10.1007/s10915-022-01784-0	journal	March 2022
Fully multidimensional flux-corrected transport algorithms for fluids Zalesak, Steven T. Journal of Computational Physics, Vol. 31, Issue 3 https://doi.org/10.1016/0021-9991(79)90051-2	journal	June 1979
Efficient implementation of essentially non-oscillatory shock-capturing schemes Shu, Chi-Wang; Osher, Stanley Journal of Computational Physics, Vol. 77, Issue 2 https://doi.org/10.1016/0021-9991(88)90177-5	journal	August 1988
Non-oscillatory central differencing for hyperbolic conservation laws Nessyahu, Haim; Tadmor, Eitan Journal of Computational Physics, Vol. 87, Issue 2 https://doi.org/10.1016/0021-9991(90)90260-8	journal	April 1990
Upwind methods for hyperbolic conservation laws with source terms Bermudez, Alfredo; Vazquez, Ma Elena Computers & Fluids, Vol. 23, Issue 8 https://doi.org/10.1016/0045-7930(94)90004-3	journal	November 1994
Exact solutions to the Riemann problem of the shallow water equations with a bottom step Alcrudo, Francisco; Benkhaldoun, Fayssal Computers & Fluids, Vol. 30, Issue 6 https://doi.org/10.1016/S0045-7930(01)00013-5	journal	July 2001
Numerical resolution of well-balanced shallow water equations with complex source terms Liang, Qiuhua; Marche, Fabien Advances in Water Resources, Vol. 32, Issue 6 https://doi.org/10.1016/j.advwatres.2009.02.010	journal	June 2009
Towards transient experimental water surfaces: A new benchmark dataset for 2D shallow water solvers Martínez-Aranda, S.; Fernández-Pato, J.; Caviedes-Voullième, D. Advances in Water Resources, Vol. 121 https://doi.org/10.1016/j.advwatres.2018.08.013	journal	November 2018
A correction technique for the dispersive effects of mass lumping for transport problems Guermond, Jean-Luc; Pasquetti, Richard Computer Methods in Applied Mechanics and Engineering, Vol. 253 https://doi.org/10.1016/j.cma.2012.08.011	journal	January 2013
A robust well-balanced model on unstructured grids for shallow water flows with wetting and drying over complex topography Hou, Jingming; Simons, Franz; Mahgoub, Mohamed Computer Methods in Applied Mechanics and Engineering, Vol. 257 https://doi.org/10.1016/j.cma.2013.01.015	journal	April 2013
Invariant domain preserving discretization-independent schemes and convex limiting for hyperbolic systems Guermond, Jean-Luc; Popov, Bojan; Tomas, Ignacio Computer Methods in Applied Mechanics and Engineering, Vol. 347 https://doi.org/10.1016/j.cma.2018.11.036	journal	April 2019
On the implementation of a robust and efficient finite element-based parallel solver for the compressible Navier–Stokes equations Guermond, Jean-Luc; Kronbichler, Martin; Maier, Matthias Computer Methods in Applied Mechanics and Engineering, Vol. 389 https://doi.org/10.1016/j.cma.2021.114250	journal	February 2022
Modeling hurricane waves and storm surge using integrally-coupled, scalable computations Dietrich, J. C.; Zijlema, M.; Westerink, J. J. Coastal Engineering, Vol. 58, Issue 1 https://doi.org/10.1016/j.coastaleng.2010.08.001	journal	January 2011
Efficient shallow water simulations on GPUs: Implementation, visualization, verification, and validation Brodtkorb, André R.; Sætra, Martin L.; Altinakar, Mustafa Computers & Fluids, Vol. 55 https://doi.org/10.1016/j.compfluid.2011.10.012	journal	February 2012
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 Delmas, Vincent; Soulaïmani, Azzedine Computer Physics Communications, Vol. 271 https://doi.org/10.1016/j.cpc.2021.108190	journal	February 2022
A well-balanced positivity preserving “second-order” scheme for shallow water flows on unstructured meshes Audusse, Emmanuel; Bristeau, Marie-Odile Journal of Computational Physics, Vol. 206, Issue 1 https://doi.org/10.1016/j.jcp.2004.12.016	journal	June 2005
Application of conservative residual distribution schemes to the solution of the shallow water equations on unstructured meshes Ricchiuto, M.; Abgrall, R.; Deconinck, H. Journal of Computational Physics, Vol. 222, Issue 1 https://doi.org/10.1016/j.jcp.2006.06.024	journal	March 2007
High-order well-balanced finite volume WENO schemes for shallow water equation with moving water Noelle, Sebastian; Xing, Yulong; Shu, Chi-Wang Journal of Computational Physics, Vol. 226, Issue 1 https://doi.org/10.1016/j.jcp.2007.03.031	journal	September 2007
On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas Gallardo, José M.; Parés, Carlos; Castro, Manuel Journal of Computational Physics, Vol. 227, Issue 1 https://doi.org/10.1016/j.jcp.2007.08.007	journal	November 2007
Exact solution of the Riemann problem for the shallow water equations with discontinuous bottom geometry Bernetti, R.; Titarev, V. A.; Toro, E. F. Journal of Computational Physics, Vol. 227, Issue 6 https://doi.org/10.1016/j.jcp.2007.11.033	journal	March 2008
Stabilized residual distribution for shallow water simulations Ricchiuto, Mario; Bollermann, Andreas Journal of Computational Physics, Vol. 228, Issue 4 https://doi.org/10.1016/j.jcp.2008.10.020	journal	March 2009
Positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations with source terms Zhang, Xiangxiong; Shu, Chi-Wang Journal of Computational Physics, Vol. 230, Issue 4 https://doi.org/10.1016/j.jcp.2010.10.036	journal	February 2011
Asymptotic preserving scheme for the shallow water equations with source terms on unstructured meshes Duran, A.; Marche, F.; Turpault, R. Journal of Computational Physics, Vol. 287 https://doi.org/10.1016/j.jcp.2015.02.007	journal	April 2015
Robust explicit relaxation technique for solving the Green-Naghdi equations Guermond, Jean-Luc; Popov, Bojan; Tovar, Eric Journal of Computational Physics, Vol. 399 https://doi.org/10.1016/j.jcp.2019.108917	journal	December 2019
Some exact solutions to the nonlinear shallow-water wave equations Thacker, William Carlisle Journal of Fluid Mechanics, Vol. 107, Issue -1 https://doi.org/10.1017/S0022112081001882	journal	June 1981
Contribution à l'étude des écoulements permanents et variables dans les canaux Serre, François La Houille Blanche, Vol. 39, Issue 6 https://doi.org/10.1051/lhb/1953058	journal	December 1953
A Well-Balanced Scheme for the Numerical Processing of Source Terms in Hyperbolic Equations Greenberg, J. M.; Leroux, A. Y. SIAM Journal on Numerical Analysis, Vol. 33, Issue 1 https://doi.org/10.1137/0733001	journal	February 1996
On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws Harten, Amiram; Lax, Peter D.; Leer, Bram van SIAM Review, Vol. 25, Issue 1 https://doi.org/10.1137/1025002	journal	January 1983
A Second-Order Maximum Principle Preserving Lagrange Finite Element Technique for Nonlinear Scalar Conservation Equations Guermond, Jean-Luc; Nazarov, Murtazo; Popov, Bojan SIAM Journal on Numerical Analysis, Vol. 52, Issue 4 https://doi.org/10.1137/130950240	journal	January 2014
Invariant Domains and First-Order Continuous Finite Element Approximation for Hyperbolic Systems Guermond, Jean-Luc; Popov, Bojan SIAM Journal on Numerical Analysis, Vol. 54, Issue 4 https://doi.org/10.1137/16M1074291	journal	January 2016
Well-Balanced Second-Order Approximation of the Shallow Water Equation with Continuous Finite Elements Azerad, Pascal; Guermond, Jean-Luc; Popov, Bojan SIAM Journal on Numerical Analysis, Vol. 55, Issue 6 https://doi.org/10.1137/17M1122463	journal	January 2017
Well-Balanced Second-Order Finite Element Approximation of the Shallow Water Equations with Friction Guermond, Jean-Luc; de Luna, Manuel Quezada; Popov, Bojan SIAM Journal on Scientific Computing, Vol. 40, Issue 6 https://doi.org/10.1137/17M1156162	journal	January 2018
Invariant-Domain-Preserving High-Order Time Stepping: I. Explicit Runge--Kutta Schemes Ern, Alexandre; Guermond, Jean-Luc SIAM Journal on Scientific Computing, Vol. 44, Issue 5 https://doi.org/10.1137/21M145793X	journal	October 2022
Strong Stability-Preserving High-Order Time Discretization Methods Gottlieb, Sigal; Shu, Chi-Wang; Tadmor, Eitan SIAM Review, Vol. 43, Issue 1 https://doi.org/10.1137/S003614450036757X	journal	January 2001
A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows Audusse, Emmanuel; Bouchut, François; Bristeau, Marie-Odile SIAM Journal on Scientific Computing, Vol. 25, Issue 6 https://doi.org/10.1137/S1064827503431090	journal	January 2004
Efficient Parallel 3D Computation of the Compressible Euler Equations with an Invariant-domain Preserving Second-order Finite-element Scheme Maier, Matthias; Kronbichler, Martin ACM Transactions on Parallel Computing, Vol. 8, Issue 3 https://doi.org/10.1145/3470637	journal	September 2021
The deal.II Library, Version 9.5 Arndt, Daniel; Bangerth, Wolfgang; Bergbauer, Maximilian Journal of Numerical Mathematics, Vol. 31, Issue 3 https://doi.org/10.1515/jnma-2023-0089	journal	August 2023
Finite Volume Evolution Galerkin Methods for the Shallow Water Equations with Dry Beds Bollermann, Andreas; Noelle, Sebastian; Lukáčová-Medvid’ová, Maria Communications in Computational Physics, Vol. 10, Issue 2 https://doi.org/10.4208/cicp.220210.020710a	journal	August 2011
A Survey of High Order Schemes for the Shallow Water Equations Shu, Y. Xing and C. -W. Journal of Mathematical Study, Vol. 47, Issue 3 https://doi.org/10.4208/jms.v47n3.14.01	journal	June 2014
A Second-Order Well-Balanced Positivity Preserving Central-Upwind Scheme for the Saint-Venant System Kurganov, Alexander; Petrova, Guergana Communications in Mathematical Sciences, Vol. 5, Issue 1 https://doi.org/10.4310/CMS.2007.v5.n1.a6	journal	January 2007
SERGHEI (SERGHEI-SWE) v1.0: a performance-portable high-performance parallel-computing shallow-water solver for hydrology and environmental hydraulics Caviedes-Voullième, Daniel; Morales-Hernández, Mario; Norman, Matthew R. Geoscientific Model Development, Vol. 16, Issue 3 https://doi.org/10.5194/gmd-16-977-2023	journal	February 2023

Similar Records

Downwinding for preserving strong stability in explicit integrating factor Runge-Kutta methods

Journal Article · 2019 · Pure and Applied Mathematics Quarterly · OSTI ID:1844914

Isherwood, Leah; Grant, Zachary J.; Gottlieb, Sigal

Implicit and Implicit–Explicit Strong Stability Preserving Runge–Kutta Methods with High Linear Order

Journal Article · 2017 · Journal of Scientific Computing · OSTI ID:1574706

Conde, Sidafa; Gottlieb, Sigal; Grant, Zachary J.; +1 more

Strong Stability Preserving Integrating Factor Two-Step Runge-Kutta Methods

Journal Article · 2019 · Journal of Scientific Computing · OSTI ID:1844913

Isherwood, Leah; Grant, Zachary J.; Gottlieb, Sigal

Related Subjects

97 MATHEMATICS AND COMPUTING
convex limiting
explicit Runge-Kutta
high-order accuracy
invariant-domain preserving
shallow water equations
well-balanced

Title: A high-order explicit Runge-Kutta approximation technique for the shallow water equations

Citation Formats

References (50)

Similar Records

Related Subjects