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A new conservative finite difference scheme for Boussinesq paradigm equation

Natalia Kolkovska, Milena Dimova (2012)

Open Mathematics

A family of nonlinear conservative finite difference schemes for the multidimensional Boussinesq Paradigm Equation is considered. A second order of convergence and a preservation of the discrete energy for this approach are proved. Existence and boundedness of the discrete solution on an appropriate time interval are established. The schemes have been numerically tested on the models of the propagation of a soliton and the interaction of two solitons. The numerical experiments demonstrate that the...

A new domain decomposition method for the compressible Euler equations

Victorita Dolean, Frédéric Nataf (2006)

ESAIM: Mathematical Modelling and Numerical Analysis

In this work we design a new domain decomposition method for the Euler equations in 2 dimensions. The starting point is the equivalence with a third order scalar equation to whom we can apply an algorithm inspired from the Robin-Robin preconditioner for the convection-diffusion equation [Achdou and Nataf, C. R. Acad. Sci. Paris Sér. I325 (1997) 1211–1216]. Afterwards we translate it into an algorithm for the initial system and prove that at the continuous level and for a decomposition into 2 sub-domains,...

A new maximality argument for a coupled fluid-structure interaction, with implications for a divergence-free finite element method

George Avalos, Matthew Dvorak (2008)

Applicationes Mathematicae

We consider a coupled PDE model of various fluid-structure interactions seen in nature. It has recently been shown by the authors [Contemp. Math. 440, 2007] that this model admits of an explicit semigroup generator representation 𝓐:D(𝓐)⊂ H → H, where H is the associated space of fluid-structure initial data. However, the argument for the maximality criterion was indirect, and did not provide for an explicit solution Φ ∈ D(𝓐) of the equation (λI-𝓐)Φ =F for given F ∈ H and λ > 0. The present...

A new mixed finite element method based on the Crank-Nicolson scheme for Burgers' equation

Xiaohui Hu, Pengzhan Huang, Xinlong Feng (2016)

Applications of Mathematics

In this paper, a new mixed finite element method is used to approximate the solution as well as the flux of the 2D Burgers’ equation. Based on this new formulation, we give the corresponding stable conforming finite element approximation for the P 0 2 - P 1 pair by using the Crank-Nicolson time-discretization scheme. Optimal error estimates are obtained. Finally, numerical experiments show the efficiency of the new mixed method and justify the theoretical results.

A new two-dimensional shallow water model including pressure effects and slow varying bottom topography

Stefania Ferrari, Fausto Saleri (2004)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The motion of an incompressible fluid confined to a shallow basin with a slightly varying bottom topography is considered. Coriolis force, surface wind and pressure stresses, together with bottom and lateral friction stresses are taken into account. We introduce appropriate scalings into a three-dimensional anisotropic eddy viscosity model; after averaging on the vertical direction and considering some asymptotic assumptions, we obtain a two-dimensional model, which approximates the three-dimensional...

A new two-dimensional Shallow Water model including pressure effects and slow varying bottom topography

Stefania Ferrari, Fausto Saleri (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The motion of an incompressible fluid confined to a shallow basin with a slightly varying bottom topography is considered. Coriolis force, surface wind and pressure stresses, together with bottom and lateral friction stresses are taken into account. We introduce appropriate scalings into a three-dimensional anisotropic eddy viscosity model; after averaging on the vertical direction and considering some asymptotic assumptions, we obtain a two-dimensional model, which approximates the three-dimensional...

A nonlinear elliptic equation with singular potential and applications to nonlinear field equations

Marino Badiale, Vieri Benci, Sergio Rolando (2007)

Journal of the European Mathematical Society

We prove the existence of cylindrical solutions to the semilinear elliptic problem − Δ u + u | y | 2 = f ( u ) , u ∈ H 1 ( ℝ N ) , u ≥ 0 , where ( y , z ) ∈ ℝ k × ℝ N − k , N > k ≥ 2 and f has a double-power behaviour, subcritical at infinity and supercritical near the origin. This result also implies the existence of solitary waves with nonvanishing angular momentum for nonlinear Schr¨odinger and Klein–Gordon equations.

A nonlinear plate control without linearization

Kenan Yildirim, Ismail Kucuk (2017)

Open Mathematics

In this paper, an optimal vibration control problem for a nonlinear plate is considered. In order to obtain the optimal control function, wellposedness and controllability of the nonlinear system is investigated. The performance index functional of the system, to be minimized by minimum level of control, is chosen as the sum of the quadratic 10 functional of the displacement. The velocity of the plate and quadratic functional of the control function is added to the performance index functional as...

A note of uniqueness on the Cauchy problem for Schrödinger or heat equations with degenerate elliptic principal parts

Hideki Takuwa (2004)

Bollettino dell'Unione Matematica Italiana

We study the local uniqueness in the Cauchy problem for Schrödinger or heat equations whose principal parts are nonnegative. We show the compact uniqueness under a weak form of pseudo convexity. This makes up for the known results under the conormal pseudo convexity given by Tataru, Hörmander, Robbiano- Zuily and L. T'Joen. Our method is based on a kind of integral transform and a weak form of Carleman estimate for degenerate elliptic operators.

A Note on an Application of the Lasota-York Fixed Point Theorem in the Turbulent Transport Problem

Tomasz Komorowski, Grzegorz Krupa (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

We study a model of motion of a passive tracer particle in a turbulent flow that is strongly mixing in time variable. In [8] we have shown that there exists a probability measure equivalent to the underlying physical probability under which the quasi-Lagrangian velocity process, i.e. the velocity of the flow observed from the vintage point of the moving particle, is stationary and ergodic. As a consequence, we proved the existence of the mean of the quasi-Lagrangian velocity, the so-called Stokes...

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