Global solutions to the shallow water system with a method of an additional argument

Алексеенко, С. Н.; Dontsova, Marina V.; Pelinovsky, Dmitry E.

doi:10.1080/00036811.2016.1208817

Cited by 9 publications

(7 citation statements)

References 12 publications

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“…Systems (1), (3) appear in various problems in natural sciences. For instance, such systems are applied in models of shallow water [1].…”

Section: Introductionmentioning

confidence: 99%

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Nonlocal Solvability of the Cauchy Problem for a System with Negative Functions of the Variable $t$

Dontsova

2023

Armen.J.Math.

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We obtain sufficient conditions for the existence and uniqueness of a local solution of the Cauchy problem for a quasilinear system with negative functions of the variable $t$ and show that the solution has the same $x$-smoothness as the initial function. We also obtain sufficient conditions for the existence and uniqueness of a nonlocal solution of the Cauchy problem for a quasilinear system with negative functions of the variable $t$.

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“…Systems (1), (3) appear in various problems in natural sciences. For instance, such systems are applied in models of shallow water [1].…”

Section: Introductionmentioning

confidence: 99%

“…By means of the additional argument method, we obtain the following extended characteristic system (see [1]- [7] for details):…”

Section: Introductionmentioning

confidence: 99%

Nonlocal Solvability of the Cauchy Problem for a System with Negative Functions of the Variable $t$

Dontsova

2023

Armen.J.Math.

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“…For a complete analysis of tsunami hydrodynamics, modeling, and forecasting, we refer the reader to Kanoglu et al [27], Pelinovsky [32], Madsen et al [29], and Synolakis and Bernard [38]. Mathematically rigorous treatment of SWE including the well-posedness and exact solutions can be found in Dobrokhotov and Tirozzi [19], Dobrokhotov et al [17,18], Alekseenko et al [1] and references therein. It is worth mentioning that IVP for SWE have been treated in Chugunov et al [7,8] using a perturbation approach.…”

Section: Introductionmentioning

confidence: 99%

“…Our approach goes as follows. 1 By the recipes discussed above, reduce the (nonlinear) SWE problem (7) to the linear system (13) with IC (16) and BC (18) and write it in matrix form…”

Section: Introductionmentioning

confidence: 99%

The Generalized Carrier–Greenspan Transform for the Shallow Water System with Arbitrary Initial and Boundary Conditions

et al. 2020

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We put forward a solution to the initial boundary value (IBV) problem for the nonlinear shallow water system in inclined channels of arbitrary cross section by means of the generalized Carrier-Greenspan hodograph transform (Rybkin et al. in J Fluid Mech, 748:416-432, 2014). Since the Carrier-Greenspan transform, while linearizing the shallow water system, seriously entangles the IBV in the hodograph plane, all previous solutions required some restrictive assumptions on the IBV conditions, e.g., zero initial velocity, smallness of boundary conditions. For arbitrary non-breaking initial conditions in the physical space, we present an explicit formula for equivalent IBV conditions in the hodograph plane, which can readily be treated by conventional methods. Our procedure, which we call the method of data projection, is based on the Taylor formula and allows us to reduce the transformed IBV data given on curves in the hodograph plane to the equivalent data on lines. Our method works equally well for any inclined bathymetry (not only plane beaches) and, moreover, is fully analytical for U-shaped bays. Numerical simulations show that our method is very robust and can be used to give express forecasting of tsunami wave inundation in narrow bays and fjords.

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“…For a complete analysis of tsunami hydrodynamics, modeling, and forecasting, we refer the reader to ; Pelinovsky (2006); Madsen et al (2008), and Synolakis and Bernard (2006). Mathematically rigorous treatment of SWE including the well-posedness and exact solutions can be found in ; Dobrokhotov et al ( , 2013, Alekseenko et al (2017) and references therein. It is worth mentioning that IVP for SWE have been treated in Chugunov et al (2014Chugunov et al ( , 2020 using a perturbation approach.…”

Section: Introductionmentioning

confidence: 99%

The Generalized Carrier-Greenspan Transform for the shallow water system with arbitrary initial and boundary conditions

Rybkin,

Nicolsky,

Pelinovsky

et al. 2020

Preprint

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We put forward a solution to the initial boundary value (IBV) problem for the nonlinear shallow water system in inclined channels of arbitrary cross-section by means of the generalized Carrier-Greenspan hodograph transform (Rybkin et al., 2014). Since the Carrier-Greenspan transform, while linearizing the shallow water system, seriously entangles the IBV in the hodograph plane, all previous solutions required some restrictive assumptions on the IBV conditions, e.g., zero initial velocity, smallness of boundary conditions. For arbitrary non-breaking initial conditions in the physical space, we present an explicit formula for equivalent IBV conditions in the hodograph plane, which can readily be treated by conventional methods. Our procedure, which we call the method of data projection, is based on the Taylor formula and allows us to reduce the transformed IBV data given on curves in the hodograph plane to the equivalent data on lines. Our method works equally well for any inclined bathymetry (not only plane beaches)

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Global solutions to the shallow water system with a method of an additional argument

Cited by 9 publications

References 12 publications

Nonlocal Solvability of the Cauchy Problem for a System with Negative Functions of the Variable $t$

Nonlocal Solvability of the Cauchy Problem for a System with Negative Functions of the Variable $t$

The Generalized Carrier–Greenspan Transform for the Shallow Water System with Arbitrary Initial and Boundary Conditions

The Generalized Carrier-Greenspan Transform for the shallow water system with arbitrary initial and boundary conditions

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