New formulation of the two-dimensional steep-slope shallow water equations. Part I: Theory and analysis

Maranzoni, Andrea; Tomirotti, Massimo

doi:10.1016/j.advwatres.2022.104255

Cited by 8 publications

(3 citation statements)

References 66 publications

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“…with the normalized potential energy density 𝑢(ℎ) defined in (5). The positive vector 𝒓 ∈ ℝ 𝑁+1 and the diagonal matrix 𝑴 = 𝜌 VT diag(𝒓) depend on the particular scheme considered.…”

Section: Energy Conserving Discretizationmentioning

confidence: 99%

“…These open-channel problems are, for example, irrigation channels with rectangular cross-sections of the river bed [1][2][3]. Another application of the shallow water equations is the modeling of snow avalanches [4,5]. In [6,7], control design for shallow water waves in a tube with a moving boundary and an arbitrary cross-section has been considered.…”

Section: Introductionmentioning

confidence: 99%

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Control‐oriented models of the shallow water equations using energy‐conserving discretization schemes

Mayer,

Wurm,

Woittennek

2023

Proc Appl Math and Mech

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In this study, we introduce higher‐order approximation schemes for a 1D shallow‐water model with a moving boundary and arbitrary cross‐section. The model equations are formulated using Lagrange coordinates to handle the time‐varying spatial domain. By discretizing the action functional on a material‐fixed grid and applying an appropriate quadrature scheme, we derive a finite‐dimensional model. This model, taking mass conservation into account as an auxiliary condition, results in a system of semi‐explicit differential‐algebraic equations (DAE). Unlike previous work, we employ higher‐order quadrature formulae to enhance numerical accuracy, albeit at the cost of more complex nonlinear DAE. In order to compare the performance of the resulting models obtained from using different quadrature schemes, a comprehensive simulation study is conducted.

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Section: Energy Conserving Discretizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Control‐oriented models of the shallow water equations using energy‐conserving discretization schemes

Mayer,

Wurm,

Woittennek

2023

Proc Appl Math and Mech

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“…Considering the 1D problem presented in Figure 1, the friction forces exerted over the bed and the pressure terms can be corrected by replacing the gravity acceleration g by g ′ = gcos 2 θ (Ni and others, 2019;Zugliani and Rosatti, 2021;Maranzoni and Tomirotti, 2022). Thus, assuming that the free surface flow is parallel to the terrain, hydrostatic pressure in the local axis z ′ is produced such that p ′ = rg cos u(h ′ − z ′ ).…”

Section: D Shallow Water Equations For Non-newtonian Flowsmentioning

confidence: 99%

Numerical modelling of dense snow avalanches with a well-balanced scheme based on the 2D shallow water equations

Sanz-Ramos¹,

Castellet²,

Oller³

et al. 2023

J. Glaciol.

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A common technique for simulating non–Newtonian fluid dynamics, such as snow avalanches, is to solve the Shallow Water Equations (SWE), together with a rheological model describing the momentum dissipation by shear stresses. Friction and cohesion terms are commonly modelled using the Voellmy friction model and, recently, the Bartelt cohesion model. Here, an adaptation of the Roe scheme that ensures the balance between the flux and pressure gradients and the friction source term is presented. An upwind scheme was used for the discretisation of the SWE numerical fluxes and the non–velocity-dependent terms of the friction–cohesion model, whereas a centred scheme was used for the velocity-dependent source terms. The model was tested in analytically solvable settings, laboratory experiments and real cases. In all cases, the model performed well, avoiding numerical instabilities and achieving stable and consistent solution even for an avalanche stopping on a sloping terrain.

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New formulation of the two-dimensional steep-slope shallow water equations. Part II: Numerical modeling, validation, and application

Maranzoni

Tomirotti

2023

Advances in Water Resources

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New formulation of the two-dimensional steep-slope shallow water equations. Part I: Theory and analysis

Cited by 8 publications

References 66 publications

Control‐oriented models of the shallow water equations using energy‐conserving discretization schemes

Control‐oriented models of the shallow water equations using energy‐conserving discretization schemes

Numerical modelling of dense snow avalanches with a well-balanced scheme based on the 2D shallow water equations

New formulation of the two-dimensional steep-slope shallow water equations. Part II: Numerical modeling, validation, and application

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