Application of conservative residual distribution schemes to the solution of the shallow water equations on unstructured meshes

Ricchiuto, Mario; Abgrall, Rémi; Deconinck, H.

doi:10.1016/j.jcp.2006.06.024

Cited by 85 publications

(110 citation statements)

References 44 publications

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“…These results can be directly compared with those in [22,24,20,17]. We observe that our results reproduce well the interaction and no oscillations are observed.…”

Section: Perturbation Of a Lake At Rest In 2dsupporting

confidence: 69%

“…This problem is taken from [20], and it involves flow over flat bed. We consider the break of a circular dam separating two basins with water levels h = 10 and h = 0.5.…”

Section: Transcritical Break Of a Circular Dammentioning

confidence: 99%

“…The simulations have been run with our numerical scheme until time t = 3 s on a uniform mesh with (200 × 200) nodes. It is useful to introduce the Froude number [20]:…”

Section: Transcritical Break Of a Circular Dammentioning

confidence: 99%

“…In this section we consider a test initially proposed in [13], and recently used in [22,24,20,17]. We show numerical results for a classical example of a small perturbation of a two-dimensional steady-state flow.…”

Section: Perturbation Of a Lake At Rest In 2dmentioning

confidence: 99%

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A new well-balanced non-oscillatory central scheme for the shallow water equations on rectangular meshes

Capilla

Balaguer-Beser

2013

Journal of Computational and Applied Mathematics

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This paper is concerned with the development of high-order well-balanced central schemes to solve the shallow water equations in two spatial dimensions. A Runge-Kutta scheme with a natural continuous extension is applied for time discretization. A Gaussian quadrature rule is used to evaluate time integrals and a three-degree polynomial which calculates point-values from cell averages or flux values by avoiding the increase in the number of solution extrema at the interior of each cell is used as reconstruction operator. That polynomial also guarantees that the number of extrema does not exceed the initial number of extrema and thus it avoids spurious numerical oscillations in the computed solution. A new procedure has been defined to evaluate the flux integrals and to approach the 2D source term integrals in order to verify the exact C-property, using the water surface elevation instead of the water depth as a variable. Numerical experiments have confirmed the high-resolution properties of our numerical scheme in 2D test problems. The well-balanced property of the resulting scheme has also been investigated.

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“…These results can be directly compared with those in [22,24,20,17]. We observe that our results reproduce well the interaction and no oscillations are observed.…”

Section: Perturbation Of a Lake At Rest In 2dsupporting

confidence: 69%

“…This problem is taken from [20], and it involves flow over flat bed. We consider the break of a circular dam separating two basins with water levels h = 10 and h = 0.5.…”

Section: Transcritical Break Of a Circular Dammentioning

confidence: 99%

“…The simulations have been run with our numerical scheme until time t = 3 s on a uniform mesh with (200 × 200) nodes. It is useful to introduce the Froude number [20]:…”

Section: Transcritical Break Of a Circular Dammentioning

confidence: 99%

Section: Perturbation Of a Lake At Rest In 2dmentioning

confidence: 99%

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A new well-balanced non-oscillatory central scheme for the shallow water equations on rectangular meshes

Capilla

Balaguer-Beser

2013

Journal of Computational and Applied Mathematics

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“…The link between (26) and the more general case of (22) is hidden in the computation of the element residuals φ K j . Very simple algebraic manipulations show that…”

Section: Nonlinear Conservation Laws and Conservationmentioning

confidence: 99%

An ALE Formulation for Explicit Runge–Kutta Residual Distribution

2014

Self Cite

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In this paper we consider the solution of hyperbolic conservation laws on moving meshes by means of an Arbitrary Lagrangian Eulerian (ALE) formulation of the Runge-Kutta Residual Distribution (RD) schemes of Ricchiuto and Abgrall (J Comput Phys 229(16):5653-5691, 2010). Up to the authors knowledge, the problem of recasting RD schemes into ALE framework has been solved with first order explicit schemes and with second order implicit schemes. Our resulting scheme is explicit and second order accurate when computing discontinuous solutions.

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Residual Distribution Schemes: Foundations and Analysis

Deconinck¹,

Ricchiuto²

2004

Encyclopedia of Computational Mechanics

101

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In this article, we introduce and analyze the class of numerical schemes known as residual distribution or fluctuation splitting schemes. The root of this family of discretizations is found mainly in works aiming at generalizing the finite volume techniques to a genuinely multidimensional upwinding context, as in, for example (Hall, Morton, Ni, Roe). However, the final result is a numerical method sharing a lot with other techniques, such as finite element schemes (Carette, Deconinck, Paillere and Roe, 1995). Our aim is to look at fluctuation splitting/residual distribution methods from a generic theoretical point of view. We discuss in a general way the basic principles and the criteria used in the design of the schemes: positivity, k‐exacteness, energy stability, well‐balancedness. We also show similarities with known, more classical, discretization techniques, as well as the main distinguishing features that make residual distribution methods a valid alternative to these classical schemes. Finally, we present some numerical applications and comparisons. Throughout the text, we give an extensive overview of the existing literature, and finally conclude the chapter by reviewing most of the ongoing research on the topic.

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Application of conservative residual distribution schemes to the solution of the shallow water equations on unstructured meshes

Cited by 85 publications

References 44 publications

A new well-balanced non-oscillatory central scheme for the shallow water equations on rectangular meshes

A new well-balanced non-oscillatory central scheme for the shallow water equations on rectangular meshes

An ALE Formulation for Explicit Runge–Kutta Residual Distribution

Residual Distribution Schemes: Foundations and Analysis

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