AENO: a Novel Reconstruction Method in Conjunction with ADER Schemes for Hyperbolic Equations

Toro, Eleuterio F.; Santacá, Andrea; Montecinos, Gino I.; Celant, Morena; Müller, Lucas O.

doi:10.1007/s42967-021-00147-0

Cited by 9 publications

(6 citation statements)

References 42 publications

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“…As expected from the settings of this test, the results for the variable h are similar to those of η and are not reported here. The AENO method reached the expected rate in all norms, being suboptimal in L ∞ norm for η, similarly to what obtained in [45] in their application on blood flows. Although converging to the correct solution, the ENO method did not reach the expected rate in all norms.…”

Section: Numerical Convergence Studysupporting

confidence: 74%

“…First we deal with the reconstruction problem. We adopt the AENO reconstruction procedure, an averaged variant of the popular ENO method [22], recently proposed by Toro et al [45]. To achieve secondorder of accuracy we need to construct first-degree polynomials p i in each cell I i at time t n from the given cell averages {Q n i } of the form…”

Section: Nonlinear Reconstruction Techniquementioning

confidence: 99%

“…ADER has also been applied to problems governed by the non-conservative SVE equations [5,4,37]. Polynomial reconstruction is performed employing the AENO reconstruction procedure, an averaged variant of the popular ENO method [22], recently proposed by Toro et al [45].…”

Section: Introductionmentioning

confidence: 99%

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A splitting scheme for the coupled Saint-Venant-Exner model

Siviglia,

Vanzo,

Toro

2021

Preprint

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We present a splitting method for the one-dimensional Saint-Venant-Exner equations used for describing the bed evolution in shallow water systems. We adapt the flux vector splitting approach of Toro and Vázquez-Cendón [42] and identify one subsystem of conservative equations (advection system) and one of non-conservative equations (pressure system), both having a very simple eigenstructure compared to the full system. The final numerical scheme is constructed using a Godunov-type path-conservative scheme for the pressure system and a simple conservative Godunov method for the advection system and solved following a coupled solution strategy. The resulting first-order accurate method is extended to second order of accuracy in space and time via the ADER approach together with an AENO reconstruction technique. Accuracy, robustness and well-balanced properties of the resulting scheme are assessed through a carefully selected suite of testcases. The scheme is exceedingly simple, accurate and robust as the sophisticated Godunov methods. A distinctive feature of the novel scheme is its flexibility in the choice of the sediment transport closure formula, which makes it particularly attractive for scientific and engineering applications.

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Section: Numerical Convergence Studysupporting

confidence: 74%

Section: Nonlinear Reconstruction Techniquementioning

confidence: 99%

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A splitting scheme for the coupled Saint-Venant-Exner model

Siviglia,

Vanzo,

Toro

2021

Preprint

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“…( 37) describes the sediment-water mass momentum conservation. The system given by ( 37) is preferred than the system established in (36). We see that erosion/deposition ex-…”

Section: Derivation Of Averaged Momentum Equationmentioning

confidence: 90%

A new class of sediment transport models: Derivation and numerical solutions

Ndengna,

Bandji,

Njifenjou

2024

Preprint

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In coastal or estuarine environments the water fluctuations withgreater length of correlations are more intense and modify the meanmotion. The literature does not provide an averaged sedimenttransport model (STM) for coastal zones that accounts for both meanand fluctuating motions over an abrupt mobile bottom. Models basedon water are commonly used to describe sediment transport, but theyare not applicable when the flow becomes turbulent. These modelsonly consider the mean motion. To account the fluctuating motion ina STM, we assume $ |u -\overline{u}|^k \leq \tau^k h, \;\; k>0$ ($h= \mathcal{O}(\varepsilon)$ is the water depth, $\lambda = const$).However, distortion no exist always since during the dam break overerodible bed, the water near the bed can become dense thus weaklyturbulent. To always ensure the turbulence existence, we assume that$\tau = \mathcal{O}(\gamma^\alpha) \;\; \varepsilon^2 \ll \gamma \ll1$ and $\rho = \mathcal{O}(\gamma)$ with $\gamma = const$.Additionally we assume a weakly concentrated approximation ($d_{50}= \mathcal{O}(\varepsilon^\alpha)$, $\delta = \mathcal{O}(\epsilon),1< \alpha <3$) to derive the model. The model obtained in thisstudy builds upon previous work by authors in 2007, 2012, and 2018,while also improving upon more recent models developed in 2022 and2023. The hyperbolic STM is subject to various complexities arisingfrom turbulence and several nonlinear coupled terms. Furthermore,solving this problem still poses a computational challenge in termsof accuracy, convergence, robustness, and efficiency. To addressthis challenge, we propose a new path-conservative method called $\text{HLL}_{**}$, which we combine with a modified Averaging method.The Essentially Non-Oscillatory (AENO) nonlinear reconstructionmethod is used to approximate the model. This method is noteworthybecause it generalizes several HLL-based schemes developed in recentyears. Numerical test cases have been performed.

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“…Furthermore, in this context, we take advantage of the dual mesh structure, and a Galerkin approach is employed for the computation of the gradients needed in the reconstruction and half-in-time evolution steps [63,64]. Further details on the classical ADER approach are presented in [65][66][67][68][69], and the modern ADER-DG methodology can be found, for instance, in [70,71]. For a more comprehensive review of the latest advances in this field, we refer to [72,73].…”

Section: Introductionmentioning

confidence: 99%

Staggered Semi-Implicit Hybrid Finite Volume/Finite Element Schemes for Turbulent and Non-Newtonian Flows

2021

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This paper presents a new family of semi-implicit hybrid finite volume/finite element schemes on edge-based staggered meshes for the numerical solution of the incompressible Reynolds-Averaged Navier–Stokes (RANS) equations in combination with the k−ε turbulence model. The rheology for calculating the laminar viscosity coefficient under consideration in this work is the one of a non-Newtonian Herschel–Bulkley (power-law) fluid with yield stress, which includes the Bingham fluid and classical Newtonian fluids as special cases. For the spatial discretization, we use edge-based staggered unstructured simplex meshes, as well as staggered non-uniform Cartesian grids. In order to get a simple and computationally efficient algorithm, we apply an operator splitting technique, where the hyperbolic convective terms of the RANS equations are discretized explicitly at the aid of a Godunov-type finite volume scheme, while the viscous parabolic terms, the elliptic pressure terms and the stiff algebraic source terms of the k−ε model are discretized implicitly. For the discretization of the elliptic pressure Poisson equation, we use classical conforming P1 and Q1 finite elements on triangles and rectangles, respectively. The implicit discretization of the viscous terms is mandatory for non-Newtonian fluids, since the apparent viscosity can tend to infinity for fluids with yield stress and certain power-law fluids. It is carried out with P1 finite elements on triangular simplex meshes and with finite volumes on rectangles. For Cartesian grids and more general orthogonal unstructured meshes, we can prove that our new scheme can preserve the positivity of k and ε. This is achieved via a special implicit discretization of the stiff algebraic relaxation source terms, using a suitable combination of the discrete evolution equations for the logarithms of k and ε. The method is applied to some classical academic benchmark problems for non-Newtonian and turbulent flows in two space dimensions, comparing the obtained numerical results with available exact or numerical reference solutions. In all cases, an excellent agreement is observed.

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AENO: a Novel Reconstruction Method in Conjunction with ADER Schemes for Hyperbolic Equations

Cited by 9 publications

References 42 publications

A splitting scheme for the coupled Saint-Venant-Exner model

A splitting scheme for the coupled Saint-Venant-Exner model

A new class of sediment transport models: Derivation and numerical solutions

Staggered Semi-Implicit Hybrid Finite Volume/Finite Element Schemes for Turbulent and Non-Newtonian Flows

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