Very high order well-balanced schemes for non-prismatic one-dimensional channels with arbitrary shape

Escalante, Cipriano; Castro, Manuel J.; Semplice, Matteo

doi:10.1016/j.amc.2021.125993

Cited by 5 publications

(4 citation statements)

References 31 publications

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“…Comparison of 1D and 2D solutions. We compare our 1D solver with the 2D shallow water solution (12). The numerical solution of ( 12) has been computed by the free and open source ToolBox FullSWOF2D (Full Shallow-Water equations for Overland Flow in 2D), which is a C++ code for simulations in two dimensions [10].…”

Section: 4mentioning

confidence: 99%

“…In this article, we propose new coupling conditions at the junction that depend on the angles with which the channels intersect at the junction allowing also for channels with different sections, [12,28]. Away from the junction we assume the solution to be 1D, while we describe the junction as a 2D region where coupling occurs between the branches.…”

Section: Introductionmentioning

confidence: 99%

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Angle dependence in coupling conditions for shallow water equations at channel junctions

Briani

Puppo

Ribot

2022

Computers & Mathematics with Applications

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Section: 4mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Angle dependence in coupling conditions for shallow water equations at channel junctions

Briani

Puppo

Ribot

2022

Computers & Mathematics with Applications

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“…Several WB numerical methods for SW models in channels have been proposed. For example, schemes capable of preserving "lake-at-rest" steady states were introduced in [2,40,42] in the case of rectangular cross-section channels and in [16,20,21] in a more general case of arbitrary cross-section channels. In [34], a relaxation scheme that can exactly preserve moving-water equilibria for rectangular cross-section channels was proposed.…”

Section: Introductionmentioning

confidence: 99%

A flux globalization based well-balanced path-conservative central-upwind scheme for the shallow water flows in channels

Chen¹,

Kurganov²,

Na³

2023

ESAIM: M2AN

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We develop a flux globalization based well-balanced (WB) path-conservative central-upwind (PCCU) scheme for the one-dimensional shallow water flows in channels. Challenges in developing numerical methods for the studied system are mainly related to the presence of nonconservative terms modeling the flow when the channel width and bottom topography are discontinuous. We use the path-conservative technique to treat these nonconservative product terms and implement this technique within the flux globalization framework, for which the friction and aforementioned nonconservative terms are incorporated into the global flux: This results in a quasi-conservative system, which is numerically solved using the Riemann-problem-solver-free central-upwind scheme. The WB property of the resulting scheme (that is, its ability to exactly preserve both still- and moving-water equilibria at the discrete level) is ensured by performing piecewise linear reconstruction for the equilibrium variables rather than the conservative variables, and then evaluating the global flux using the obtained point values of the equilibrium quantities. The robustness and excellent performance of the proposed flux globalization based WB PCCU scheme are demonstrated in several numerical examples with both continuous and discontinuous channel width and bottom topography. In these examples, we clearly demonstrate the advantage of the proposed scheme over its simpler counterparts.

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“…Achieving this well-balanced property in the context of Godunov-type schemes is non-trivial and a lot of research has been carried out in this direction in the last decades. Recent developments concerning well-balanced Godunov-type finite volume schemes for the Saint-Venant equations can be found, for example, in [92,145,85,108,91,71,132,124,125,85,28,3,18,54,137,119,55,107,75,57] while modern work on multilayer models is presented in [64,93,110]. A review of classical shock-capturing finite volume schemes for shallow water flows is available in the textbook [142].…”

Section: Introductionmentioning

confidence: 99%

A staggered semi-implicit hybrid finite volume / finite element scheme for the shallow water equations at all Froude numbers

Busto,

Dumbser

2023

Preprint

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We present a novel staggered semi-implicit hybrid finite volume / finite element method for the numerical solution of the shallow water equations at all Froude numbers on unstructured meshes. A semi-discretization in time of the conservative Saint-Venant equations with bottom friction terms leads to its decomposition into a first order hyperbolic subsystem containing the nonlinear convective term and a second order wave equation for the pressure. For the spatial discretization of the free surface elevation an unstructured mesh composed of triangular simplex elements is considered, whereas a dual grid of the edge-type is employed for the computation of the depth-averaged momentum vector. The first stage of the proposed algorithm consists in the solution of the nonlinear convective subsystem using an explicit Godunov-type finite volume method on the staggered dual grid. Next, a classical continuous finite element scheme provides the free surface elevation at the vertices of the primal simplex mesh. The semi-implicit strategy followed in this paper circumvents the contribution of the surface wave celerity to the CFL-type time step restriction, hence making the proposed algorithm well-suited for the solution of low Froude number flows. It can be shown that in the low Froude number limit the proposed algorithm reduces to a semi-implicit hybrid FV/FE projection method for the incompressible Navier-Stokes equations. At the same time, the conservative formulation of the governing equations also allows the discretization of high Froude number flows with shock waves. As such, the new hybrid FV/FE scheme can be considered an all-Froude number solver, able to deal simultaneously with both, subcritical as well as supercritical flows. Besides, the algorithm is well balanced by construction. The accuracy of the overall methodology is studied numerically and the C-property is proven theoretically and is also validated via numerical experiments. The numerical solution of several Riemann problems, including flat bottom and a bottom with jumps, attests the robustness of the new method to deal also with flows containing bores and discontinuities. Finally, a 3D dam break problem over a dry bottom is studied and our numerical results are successfully compared with numerical reference solutions obtained for the 3D free surface Navier-Stokes equations and with available experimental reference data.

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Very high order well-balanced schemes for non-prismatic one-dimensional channels with arbitrary shape

Cited by 5 publications

References 31 publications

Angle dependence in coupling conditions for shallow water equations at channel junctions

Angle dependence in coupling conditions for shallow water equations at channel junctions

A flux globalization based well-balanced path-conservative central-upwind scheme for the shallow water flows in channels

A staggered semi-implicit hybrid finite volume / finite element scheme for the shallow water equations at all Froude numbers

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