Some considerations on numerical schemes for treating hyperbolicity issues in two-layer models

Sarno, Luca; Carravetta, Armando; Martino, Riccardo; Papa, Maria Nicolina; Tai, Yih-Chin

doi:10.1016/j.advwatres.2016.12.014

Cited by 21 publications

(22 citation statements)

References 38 publications

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“…These conditions are found in some stratified flows in nature, such as exchange flows through sea straits Chakir et al, 2009) or some cases of highly stratified estuaries Krvavica et al (2017b). For a general application, however, this condition may not be necessary to ensure hyperbolicity, as demonstrated by Sarno et al (2017).…”

Section: Governing System Of Equationsmentioning

confidence: 99%

“…Since explicit eigenvalues of a two-layer system are considered too complicated and unavailable (Pelanti et al, 2008;Kim and LeVeque, 2008;Abgrall and Karni, 2009;Fernández-Nieto et al, 2011;Fjordholm, 2012;Sarno et al, 2017), the following approximations derived under the assumption of r ≈ 1 and u 1 ≈ u 2 are usually used for internal and external eigenvalues (Schijf and Schönfled, 1953):…”

Section: Governing System Of Equationsmentioning

confidence: 99%

Section: Numerical Treatment For the Loss Of Hyperbolicitymentioning

confidence: 99%

“…where p is derivative of polynomial p, and R(p, p ) is the Sylvester matrix of p and p (Sarno et al, 2017). For a quartic equation, this formula yields a rather long expression (for details see (Sarno et al, 2017)). In this work, a similar approach to Sarno et al (2017) is proposed; however, the choice of the discriminant and the implementation of the hyperbolicity correction differs.…”

Section: Numerical Treatment For the Loss Of Hyperbolicitymentioning

confidence: 99%

“…Because of the coupling and the corresponding 4 × 4 flux matrix, some authors suggest that it is not possible to derive the explicit form of eigenvalues, e.g., "...simple explicit expressions of the system's eigenvalues cannot be derived..." (Pelanti et al, 2008), "...the explicit expression for the eigenvalues cannot be found." (Kim and LeVeque, 2008), "The coupling between the layers... does not provide explicit access to the system eigenstructure" (Abgrall and Karni, 2009), whereas others are aware of the existence of the analytical solutions to quartic equations but considered them to be too complicated or less efficient, e.g., "...there is not an easy explicit expression of the eigenvalues..." , "...a direct calculation of its eigenvalues can be hard..." (Fjordholm, 2012), "...a closed form of the eigenvalues is non-trivial..." (Sarno et al, 2017), etc. On the other hand, Cardano-Vieta formula for cubic equations has been used as a more efficient approach in comparison to numerical solvers when computing eigenstructure of Saint Venant-Exner models, defined by a cubic characteristic equation (see Castro et al (2009) and Carraro et al (2018)).…”

Section: Introductionmentioning

confidence: 99%

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Analytical implementation of Roe solver for two-layer shallow water equations with accurate treatment for loss of hyperbolicity

Krvavica

Tuhtan

Jelenić

2018

Advances in Water Resources

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A new implementation of the Roe scheme for solving two-layer shallow-water equations is presented in this paper. The proposed A-Roe scheme is based on the analytical solution to the characteristic quartic of the flux matrix, which is an efficient alternative to a numerical eigensolver. Additionally, an accurate method for maintaining the hyperbolic character of the governing system is proposed. The efficiency of the quartic closed-form solver is examined and compared to numerical eigensolvers. Furthermore, the accuracy and computational speed of the A-Roe scheme is compared to the Roe, Lax-Friedrichs, GFORCE, PVM, and IFCP schemes. Finally, numerical tests are presented to evaluate the efficiency of the iterative treatment for the hyperbolicity loss. The proposed A-Roe scheme is as accurate as the Roe scheme, but much faster, with computational speeds closer to the GFORCE and IFCP scheme.

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Section: Governing System Of Equationsmentioning

confidence: 99%

Section: Governing System Of Equationsmentioning

confidence: 99%

Section: Numerical Treatment For the Loss Of Hyperbolicitymentioning

confidence: 99%

Section: Numerical Treatment For the Loss Of Hyperbolicitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Analytical implementation of Roe solver for two-layer shallow water equations with accurate treatment for loss of hyperbolicity

Krvavica

Tuhtan

Jelenić

2018

Advances in Water Resources

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Kinematics of flow mass movements on inclined surfaces

Rendina

Viccione

Cascini

2019

Theor. Comput. Fluid Dyn.

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Experimental Investigation on the Effects of the Fixed Boundaries in Channelized Dry Granular Flows

et al. 2017

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The dynamics of granular mixtures, involved in several geophysical phenomena like rock avalanches and debris flows, is far from being completely understood. Several features of their motion, such as non-local and boundary effects, still represent open problems. An extensive experimental study on free-surface channelized granular flows is here presented, where the effects of the fixed boundaries are systematically investigated. The entire experimental data set is obtained by using a homogenous acetal-polymeric granular material and three different basal surfaces, allowing different kinematic boundary conditions. Velocity profiles at both the sidewall and the free surface are obtained by using high-speed cameras and the open-source particle image velocimetry code, PIVlab. Significantly, different sidewall velocity profiles are observed by varying the basal roughness and the flow depth. Owing to sidewall friction and non-local effects, such profiles exhibit a clear rheological stratification for high enough flow depths and they can be well described by recurring to composite functions, variously formed of linear, Bagnold and exponential scalings. Moreover, it has been discovered that transitions from one velocity profile to another are also possible on the same basal surface by merely varying the flow depth. This shape transition is due partly to the sidewall resistances, the basal boundary condition and, in particular, the occurrence/inhibition of basal grain rolling. In most of the experiments, the normal-to-bed velocity profiles and the velocity measurements at the free surface strongly suggest the occurrence of a secondary circulating flow, made possible by a chiefly collisional regime beneath the free surface

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Some considerations on numerical schemes for treating hyperbolicity issues in two-layer models

Cited by 21 publications

References 38 publications

Analytical implementation of Roe solver for two-layer shallow water equations with accurate treatment for loss of hyperbolicity

Analytical implementation of Roe solver for two-layer shallow water equations with accurate treatment for loss of hyperbolicity

Kinematics of flow mass movements on inclined surfaces

Experimental Investigation on the Effects of the Fixed Boundaries in Channelized Dry Granular Flows

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