Paralinearization of the Dirichlet to Neumann Operator, and Regularity of Three-Dimensional Water Waves

When both gravity and surface tension effects are present, surface solitary water waves are known to exist in both two- and three-dimensional infinitely deep fluids. We describe here solutions bridging these two cases: travelling waves which are localized in the propagation direction and periodic in the transverse direction. These transversally periodic gravity–capillary solitary waves are found to be of either elevation or depression type, tend to plane waves below a critical transverse period and tend to solitary lumps as the transverse period tends to infinity. The waves are found numerically in a Hamiltonian system for water waves simplified by a cubic truncation of the Dirichlet-to-Neumann operator. This approximation has been proved to be very accurate for both two- and three-dimensional computations of fully localized gravity–capillary solitary waves. The stability properties of these waves are then investigated via the time evolution of perturbed wave profiles.

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Nonlinear irrotational water waves over variable bathymetry. The Hamiltonian approach with a new efficient representation of the dirichlet to neumann operator

2015 Days on Diffraction (DD) ◽

10.1109/dd.2015.7354825 ◽

2015 ◽

Author(s):

Gerassimos A. Athanassoulis ◽

Christos E. Papoutsellis

Keyword(s):

Water Waves ◽

Hamiltonian Approach ◽

Neumann Operator ◽

Efficient Representation ◽

Dirichlet To Neumann Operator

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A three-dimensional Dirichlet-to-Neumann operator for water waves over topography

Journal of Fluid Mechanics ◽

10.1017/jfm.2018.241 ◽

2018 ◽

Vol 845 ◽

pp. 321-345 ◽

Cited By ~ 4

Author(s):

D. Andrade ◽

A. Nachbin

Keyword(s):

Free Surface ◽

Water Waves ◽

Three Dimensional ◽

Matrix Decomposition ◽

Fourier Integral ◽

Two Dimensional ◽

Accurate Performance ◽

Surface Water Waves ◽

Dimensional Simulation ◽

Dirichlet To Neumann Operator

Surface water waves are considered propagating over highly variable non-smooth topographies. For this three-dimensional problem a Dirichlet-to-Neumann (DtN) operator is constructed reducing the numerical modelling and evolution to the two-dimensional free surface. The corresponding discrete Fourier integral operator is defined through a matrix decomposition. The topographic component of the decomposition requires special care, and a Galerkin method is provided accordingly. One-dimensional numerical simulations, along the free surface, validate the DtN formulation in the presence of a large-amplitude rapidly varying topography. An alternative conformal-mapping-based method is used for benchmarking. A two-dimensional simulation in the presence of a Luneburg lens (a particular submerged mound) illustrates the accurate performance of the three-dimensional DtN operator.

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Analytic semigroups generated by Dirichlet-to-Neumann operators on manifolds

Semigroup Forum ◽

10.1007/s00233-021-10192-z ◽

2021 ◽

Author(s):

Tim Binz

Keyword(s):

Boundary Conditions ◽

Elliptic Operator ◽

Compact Manifold ◽

Analytic Semigroup ◽

Continuous Functions ◽

Analytic Semigroups ◽

Abstract Theory ◽

Neumann Operator ◽

Wentzell Boundary Conditions ◽

Dirichlet To Neumann Operator

AbstractWe consider the Dirichlet-to-Neumann operator associated to a strictly elliptic operator on the space $$\mathrm {C}(\partial M)$$ C ( ∂ M ) of continuous functions on the boundary $$\partial M$$ ∂ M of a compact manifold $$\overline{M}$$ M ¯ with boundary. We prove that it generates an analytic semigroup of angle $$\frac{\pi }{2}$$ π 2 , generalizing and improving a result of Escher with a new proof. Combined with the abstract theory of operators with Wentzell boundary conditions developed by Engel and the author, this yields that the corresponding strictly elliptic operator with Wentzell boundary conditions generates a compact and analytic semigroups of angle $$\frac{\pi }{2}$$ π 2 on the space $$\mathrm {C}(\overline{M})$$ C ( M ¯ ) .

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