Flux Globalization Based Well-Balanced Path-Conservative Central-Upwind Schemes for Shallow Water Models

Cao, Yangyang; Kurganov, Alexander; Liu, Yongle; Xin, Ruixiao

doi:10.1007/s10915-022-01912-w

Cited by 14 publications

(13 citation statements)

References 45 publications

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“…In this section, we follow the approach recently introduced in [5,29] and introduce the flux globalization based semi-discrete WB PCCU scheme, respectively.…”

Section: Flux Globalization Based Pccu Schemementioning

confidence: 99%

“…In this paper, we develop WB schemes for the system (1.1) using a flux globalization approach which was proposed in [12] and then applied to a variety of hyperbolic systems of balance laws in [5,9,11,13,28,29,35]. In this approach, we incorporate the source terms into the fluxes and rewrite (1.1) in the following quasi-conservative form:…”

Section: Introductionmentioning

confidence: 99%

“…The drawback of the PCCU schemes, however, is that they can preserve "lake-at-rest" steady states only. In order to design moving-water equilibrium preserving PCCU schemes, the PC approach was incorporated into the flux globalization based WB CU schemes in [29]; also see [5]. This results in a flux globalization based WB PCCU scheme, which we apply to the studied SW in channel system.…”

Section: Introductionmentioning

confidence: 99%

“…5) is satisfied (and thus the reconstructed point values 𝐾 ± will take the same value). In fact, one has from (C.1), (C.3), and 𝑄 𝑗 ≡ 0 that𝐾 𝑗+1 − 𝐾 𝑗 We then note that (C.5) implies 𝑍 𝑗+1 − 𝑍 𝑗 = 𝐴 𝑗 𝜂 𝑗 − 𝐴 𝑗+1 𝜂 𝑗+1 , which together with (C.2) yields 𝑍 𝑗+1 − 𝑍 𝑗 + 𝑍 𝑗 − 𝑍 𝑗+ 1 ̂︀ 𝑤 − 𝑍 𝑗+1 ) − 𝐴 𝑗 ( ̂︀ 𝑤 − 𝑍 𝑗 ) + 𝐴 𝑗+1 (𝑍 𝑗+1 − 𝑍 𝑗 ) − (︀ 𝐴 𝑗+1 − 𝐴 𝑗 )︀ ( ̂︀ 𝑤 − 𝑍 𝑗 ) ]︀ = 0,which indicates that the "lake-at-rest" state (C.5) is preserved.…”

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confidence: 96%

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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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“…In this section, we follow the approach recently introduced in [5,29] and introduce the flux globalization based semi-discrete WB PCCU scheme, respectively.…”

Section: Flux Globalization Based Pccu Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 96%

See 2 more Smart Citations

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

Chen¹,

Kurganov²,

Na³

2023

ESAIM: M2AN

Self Cite

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“…Most recently, Cao et al. [52] constructed a flux‐globalization‐based well‐balanced path‐conservative central upwind scheme and tested the proposed formulation on 1D SWEs, including Saint–Venant systems with/without friction and rotating SWEs.…”

Section: Introductionmentioning

confidence: 99%

SUPG formulation augmented with YZβ shock‐capturing for computing shallow‐water equations

Cengizci

Uğur

2023

Z Angew Math Mech

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We demonstrate that the streamline-upwind/Petrov-Galerkin (SUPG) formulation enhanced with YZ𝛽 discontinuity-capturing, that is, the SUPG-YZ𝛽 formulation, is an efficient and robust method for computing 2D shallow-water equations (SWEs). The SUPG-stabilized semi-discrete formulation is discretized in time by employing the backward Euler time-integration scheme. The nonlinear equation systems arising from the space and time discretizations are handled using the Newton-Raphson (N-R) method at each time step. The resulting linear equation systems are solved directly at each nonlinear iteration. Two challenging test problems are provided to examine the performance of the proposed formulation and techniques. To that end, we consider a full dam-break and a partial dam-break problem. We develop the solvers in the FEniCS environment. Test computations reveal that the SUPG-YZ𝛽 formulation successfully eliminates spurious oscillations that cannot be captured with the SUPG-stabilized formulation alone in narrow regions where steep gradients occur. INTRODUCTIONMany real-world phenomena, including water flows in open channels, the transport of chemical species, flood flows (including those caused by tsunami waves, tidal flows, and storm surges), dam-break problems, and turbulence phenomena in the atmosphere and oceans can be mathematically modeled using the shallow-water equations (SWEs). Finite element methods (FEMs) are well-suited for simulating such models since they typically have complex flow domains. The flexible nature of the FEM is essential for real-world computations since we might need to work on extremely complicated geometries, for example, tidal waves or tsunamis that could happen in a zigzagging bay. A good number of studies based on various FEMs, as well as finite difference methods (FDM) and finite volume methods (FVMs), have been reported in the literature for computing SWEs. Although it is not possible to mention all of them, the following paragraphs attempt to provide a comprehensive review of featured research for computing SWEs with a particular emphasis on FEMs. For more details, the interested reader can refer to the materials provided in these studies.The authors of Refs. [1, 2] employed the standard Galerkin FEM (GFEM) with a two-step time-integration scheme they called the "selective lumping scheme" for computing SWEs. For computing two-layer SWEs, Hua and Thomasset [3] proposed a semi-implicit time-integration scheme paired with triangular elements, where the velocity components are approximated with nonconforming linear elements, and the (water) elevation is approximated with linear conforming elements. In Nessyahu and Tadmor [4], the authors proposed a second-order central difference scheme for solving nonlinear systems of hyperbolic conservation laws and evaluated the performance of the proposed formulation on Riemann-type problems. Bermudez et al. used a Lagrange-Galerkin FEM to upwind the convection term in Bermudez et al. [5]. They performed their computations on a 1D test problem and provid...

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A moving water equilibria preserving nonstaggered central scheme achieved via flux globalization for the Ripa model

Li,

Liu,

2024

Acta Math Sci

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Flux Globalization Based Well-Balanced Path-Conservative Central-Upwind Schemes for Shallow Water Models

Cited by 14 publications

References 45 publications

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

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

SUPG formulation augmented with YZβ shock‐capturing for computing shallow‐water equations

A moving water equilibria preserving nonstaggered central scheme achieved via flux globalization for the Ripa model

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