Well-Balancing via Flux Globalization: Applications to Shallow Water Equations with Wet/Dry Fronts

Chertock, Alina; Kurganov, Alexander; Liu, Xin; Liu, Yongle; Wu, Tong

doi:10.1007/s10915-021-01680-z

Cited by 18 publications

(5 citation statements)

References 23 publications

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“…The main idea here is to follow [9,[11][12][13]28] and perform the generalized minmod reconstruction of the equilibrium variables 𝑄 and 𝐾 rather than 𝑄 and 𝐸. In order to do this, we first need to compute the values of 𝐾 at the cell centers 𝑥 = 𝑥 𝑗 .…”

Section: Appendix C Flux Globalization Based Wb Cu 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%

“…This, however, will not lead to a WB scheme as 𝐴 is not an equilibrium variable, that is, it does not remain constant at steady states. We may follow the lines of [9,[11][12][13]28] and reconstruct the equilibrium quantities 𝑄 and 𝐾. This will lead to a WB CU scheme, which is expected to be highly accurate and robust as long as the function 𝜂(𝑥) and 𝑍(𝑥) are continuous.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Appendix C Flux Globalization Based Wb Cu Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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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“…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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“…It is then interesting to build schemes that are able to preserve also these equilibria at the discrete level. A class of schemes has been built in the recent years with the aim of preserving such equilibria [11,10,12]. They rely on the idea of defining a global flux that incorporates both the flux and the source term thanks to a discretization of the integral of the source term.…”

Section: Introductionmentioning

confidence: 99%

Arbitrary High Order WENO Finite Volume Scheme with Flux Globalization for Moving Equilibria Preservation

Ciallella¹,

Torlo²,

Ricchiuto³

2022

Preprint

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In the context of preserving stationary states, e.g. lake at rest and moving equilibria, a new formulation of the shallow water system, called Flux Globalization has been introduced by Cheng et al. (2019). This approach consists in including the integral of the source term in the global flux and reconstructing the new global flux rather than the conservative variables. The resulting scheme is able to preserve a large family of smooth and discontinuous steady state moving equilibria. In this work, we focus on an arbitrary high order WENO Finite Volume (FV) generalization of the global flux approach. The most delicate aspect of the algorithm is the appropriate definition of the source flux (integral of the source term) and the quadrature strategy used to match it with the WENO reconstruction of the hyperbolic flux. When this construction is correctly done, one can show that the resulting WENO FV scheme admits exact discrete steady states characterized by constant global fluxes. We also show that, by an appropriate quadrature strategy for the source, we can embed exactly some particular steady states, e.g. the lake at rest for the shallow water equations. It can be shown that an exact approximation of global fluxes leads to a scheme with better convergence properties and improved solutions. The novel method has been tested and validated on classical cases: subcritical, supercritical and transcritical flows.

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Well-Balancing via Flux Globalization: Applications to Shallow Water Equations with Wet/Dry Fronts

Cited by 18 publications

References 23 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

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

Arbitrary High Order WENO Finite Volume Scheme with Flux Globalization for Moving Equilibria Preservation

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