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The vortex method is a common numerical and theoretical approach used to implement the motion of an ideal flow, in which the vorticity is approximated by a sum of point vortices, so that the Euler equations read as a system of ordinary differential equations. Such a method is well justified in the full plane, thanks to the explicit representation formulas of Biot and Savart. In an exterior domain, we also replace the impermeable boundary by a collection of point vortices generating the circulation...
We introduce and investigate the well-posedness of a model describing the self-propelled motion of a small abstract swimmer in the 3-D incompressible fluid governed by the nonstationary Stokes equation, typically associated with low Reynolds numbers. It is assumed that the swimmer's body consists of finitely many subsequently connected parts, identified with the fluid they occupy, linked by rotational and elastic Hooke forces. Models like this are of interest in biological and engineering applications...
When applied to the linear advection problem in dimension two, the
upwind finite volume method is a non consistent scheme in the finite
differences sense but a convergent scheme. According to our previous
paper [Bouche et al.,
SIAM J. Numer. Anal.43 (2005) 578–603], a sufficient condition in order to
complete the mathematical analysis of the finite volume scheme
consists in obtaining an estimation of order p, less or equal to
one, of a quantity that depends only on the mesh and on the advection
...
We describe several results obtained recently on stochastic nonlinear Schrödinger equations. We show that under suitable smoothness assumptions on the noise, the nonlinear Schrödinger perturbed by an additive or multiplicative noise is well posed under similar assumptions on the nonlinear term as in the deterministic theory. Then, we restrict our attention to the case of a focusing nonlinearity with critical or supercritical exponent. If the noise is additive, smooth in space and non degenerate,...
In this paper, we study a Zakharov system coupled to an electron diffusion equation in order to describe laser-plasma interactions. Starting from the Vlasov-Maxwell system, we derive a nonlinear Schrödinger like system which takes into account the energy exchanged between the plasma waves and the electrons via Landau damping. Two existence theorems are established in a subsonic regime. Using a time-splitting, spectral discretizations for the Zakharov system and a finite difference scheme for the...
In this paper, we study a Zakharov system coupled to an electron
diffusion equation in order to describe laser-plasma interactions. Starting from
the Vlasov-Maxwell system, we derive a nonlinear Schrödinger
like system which takes into account the energy exchanged between the plasma waves and the electrons
via Landau damping. Two existence theorems are established in a subsonic regime.
Using a time-splitting, spectral discretizations for the Zakharov system and a
finite difference scheme for...
A computer-aided method for accurately carrying out the Chapman-Enskog expansion of the Boltzmann equation, including its inelastic variant, is presented and employed to derive a hydrodynamic description of a dilute binary mixture of smooth inelastic spheres. Constitutive relations, formally valid for all physical values of the coefficients of restitution, are calculated by carrying out the pertinent Chapman-Enskog expansion to sufficient high orders in the Sonine polynomials to ensure numerical...
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