In an earlier paper the authors introduced a new version of vortex method for three-dimensional, incompressible flows and proved that it converges to arbitrarily high order accuracy, provided we assume consistency discrete approximation Biot-Savart Law. We prove this statement here, also derive substantially sharper results two-dimensional flows. A complete, simplified proof convergence in two dimensions is included.

Recently several different approaches have been developed for the simulation of three-dimensional incompressible fluid flows using vortex methods. Some versions use detailed tracking filament structures and often local curvatures these filaments, while other methods require only crude information, such as blobs two-dimensional case. Can "crude" algorithms accurately account stretching converge? We answer this question affirmatively by constructing a new class then proving that are stable...

Viscous splitting algorithms are the underlying design principle for many numerical which solve Navier-Stokes equations at high Reynolds number. In this work, error estimates developed uniform in viscosity <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="nu"> <mml:semantics> <mml:mi>ν<!-- ν --></mml:mi> <mml:annotation encoding="application/x-tex">\nu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as it becomes...

Abstract We consider the motion of a two‐dimensional interface separating an inviscid, incompressible, irrotational fluid, influenced by gravity, from region zero density. show that under certain conditions equations motion, linearized about presumed time‐dependent solution, are wellposed; is, linear disturbances have bounded rate growth. If surface tension is neglected, well‐posed provided underlying exact satisfies condition on acceleration relative to similar criterion formulated G. I....

The well-known quasigeostrophic system (QGS) for zero Rossby number flow has been used extensively in oceanography and meteorology modeling forecasting mid-latitude oceanic atmospheric circulation. Formulation of QGS requires a (singular) perturbation expansion set primitive equations at small number, the equation expresses conservation zero-order potential vorticity flow. formal is justified by investigating behavior solutions (PE) with particular scaling, limit number. This model...

Abstract We prove the existence of solitary water waves elevation, as exact solutions equations steady inviscid flow, taking into account effect surface tension on free surface. In contrast to case without tension, a resonance occurs with periodic same speed. The wave form consists single crest elongated scale much smaller oscillation at infinity physical scale. have not proved that amplitude is actually nonzero; formal calculation suggests it exponentially small.

We develop a method for computing nearly singular integral, such as double layer potential due to sources on curve in the plane, evaluated at point near curve. The approach is regularize singularity and obtain preliminary value from standard quadrature rule. Then we add corrections errors smoothing discretization, which are found by asymptotic analysis. prove an error estimate corrected value, uniform with respect thepoint of evaluation. One application simple solving Dirichlet problem...

Abstract We present a simple, accurate method for computing singular or nearly integrals on smooth, closed surface, such as layer potentials harmonic functions evaluated at points near the surface. The integral is computed with regularized kernel and corrections are added regularization discretization, which found from analysis point. surface new quadrature rule using project onto grid in coordinate planes. does not require charts special treatment of singularity other than corrections....

We prove nonlinear stability and convergence of certain boundary integral methods for time-dependent water waves in a two-dimensional, inviscid, irrotational, incompressible fluid, with or without surface tension. The are convergent as long the underlying solution remains fairly regular (and sign condition holds case tension). Thus, numerical instabilities ruled out even fully regime. analysis is based on delicate energy estimates, following framework previously developed continuous [Beale,...

On etablit la convergence d'une methode de tourbillons pour un ecoulement tridimensionnel incompressible non visqueux sans frontieres

We develop a simple, efficient numerical method of boundary integral type for solving an elliptic partial differential equation in three-dimensional region using the classical formulation potential theory. Accurate values can be found near special corrections to standard quadrature. treat Dirichlet problem harmonic function with prescribed value bounded smooth boundary. The solution is double layer potential, whose strength by second kind. surface represented rectangular grids overlapping...