An Arbitrary High Order Well-Balanced ADER-DG Numerical Scheme for the Multilayer Shallow-Water Model with Variable Density

Fernández, Ernesto Guerrero; Díáz, Manuel; Dumbser, Michael; Luna, Tomás Morales de

doi:10.1007/s10915-021-01734-2

Cited by 12 publications

(4 citation statements)

References 80 publications

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“…It worth mentioning [20,15] where DG schemes, solving different NSW system of equations, have been developed and combined, similarly to [17], with an a posteriori sub-cell correction based a robust lower order scheme. Let us emphasize that, while these stabilization procedures are indeed close related, the one presented in this article is local to the subcell, and thus enables a better preservation of the very precise subcell resolution of DG scheme, see [32] for a comparison between the two.…”

Section: Introductionmentioning

confidence: 99%

Free-Boundary Problems for Wave–Structure Interactions in Shallow-Water: DG-ALE Description and Local Subcell Correction

Haidar,

Marche,

Vilar

2024

J Sci Comput

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We introduce a robust numerical strategy for the numerical simulation of several free-boundary problems arising in the study of nonlinear wave-structure interactions in shallow-water flows. We investigate two types of boundary-evolution equations: (i) a kinematic-type equation, associated with the interaction of waves with a moving lateral wall, (ii) a fully-nonlinear singular equation modeling the evolution of the interface between a solid obstacle placed on the surface and the fluid. At the continuous level, the flow is globally modeled with the hyperbolic Nonlinear Shallow-Water (NSW) equations, including varying topography, and at the discrete level, an arbitrary-order discontinuous Galerkin (DG) method is stabilized with a Local Subcell Correction (LSC) method. Mimicking the theoretical study of the continuous problem, suitable diffeomorphisms are introduced to recast the moving-boundary problems into fixed-boundary ones, and to compute the boundary's evolution through an Arbitrary-Lagrangian-Eulerian (ALE) description. For any order of polynomial approximation, the resulting global algorithm is shown to: (i) preserve the Discrete Geometric Conservation Law (DGCL), (ii) ensure the preservation of the water height positivity at the sub-cell level, (iii) preserve the class of motionless steady-states (well-balancing), possibly with the occurrence of a partly immersed obstacle. Several numerical computations highlight that the proposed strategy: (i) effectively approximate the new free-boundary IBVPs introduced in [19], (ii) is able to accurately handle strong flow singularities without any robustness issues, (iii) retains the highly accurate subcell resolution of DG schemes.

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Section: Introductionmentioning

confidence: 99%

Free-Boundary Problems for Wave–Structure Interactions in Shallow-Water: DG-ALE Description and Local Subcell Correction

Haidar,

Marche,

Vilar

2024

J Sci Comput

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“…[50], and a discontinuous Galerkin method was used for discretization of a variable pressure multilayer shallow‐water model in Fernández et al. [51]. Most recently, Cao et al.…”

Section: Introductionmentioning

confidence: 99%

“…In recent years, the authors proposed a staggered semi-implicit hybrid finite volume/FEM for solving SWEs at all Froude numbers on unstructured meshes in Busto and Dumbser [48], a stabilized FEM formulation is employed with the finite increment calculus (FIC) procedure in Masó et al [49], a well-balanced discontinuous Galerkin discretization of SWEs in spherical geometries was considered for oceanographic applications in Arpaia et al [50], and a discontinuous Galerkin method was used for discretization of a variable pressure multilayer shallow-water model in Fernández et al [51]. 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.…”

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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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“…The DG and ADER-DG methods have been successfully applied to geophysical flows. In the framework of the one-layer shallow water model, we highlight the work in [68][69][70]73] among others, and in the multilayer shallow water case in [74,75]. More recently, advances on hyperbolic reformulations of nonlinear dispersive shallow water systems that make use of the DG or ADER-DG techniques are found on [67,[76][77][78].…”

Section: Introductionmentioning

confidence: 99%

Well-Balanced High-Order Discontinuous Galerkin Methods for Systems of Balance Laws

2021

Self Cite

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This work introduces a general strategy to develop well-balanced high-order Discontinuous Galerkin (DG) numerical schemes for systems of balance laws. The essence of our approach is a local projection step that guarantees the exactly well-balanced character of the resulting numerical method for smooth stationary solutions. The strategy can be adapted to some well-known different time marching DG discretisations. Particularly, in this article, Runge–Kutta DG and ADER DG methods are studied. Additionally, a limiting procedure based on a modified WENO approach is described to deal with the spurious oscillations generated in the presence of non-smooth solutions, keeping the well-balanced properties of the scheme intact. The resulting numerical method is then exactly well-balanced and high-order in space and time for smooth solutions. Finally, some numerical results are depicted using different systems of balance laws to show the performance of the introduced numerical strategy.

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An Arbitrary High Order Well-Balanced ADER-DG Numerical Scheme for the Multilayer Shallow-Water Model with Variable Density

Cited by 12 publications

References 80 publications

Free-Boundary Problems for Wave–Structure Interactions in Shallow-Water: DG-ALE Description and Local Subcell Correction

Free-Boundary Problems for Wave–Structure Interactions in Shallow-Water: DG-ALE Description and Local Subcell Correction

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

Well-Balanced High-Order Discontinuous Galerkin Methods for Systems of Balance Laws

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