The Convergence of Contour Dynamics Methods

In this paper the properties of contour dynamics methods of two-dimensional incompressible inviscid vortex flows are investigated. The error estimates and the convergence of the methods for piecewise constant vorticity patches using Euler's method are obtained.     

The Evolution of Initial Small Disturbance in Discrete Computation of Contour Dynamics

In this paper, we mainly discuss the evolution of initial small disturbance in discrete computation of the contour dynamics method. For one class of smooth contour, we prove the stability of evolution of initial small disturbance based on the analysis of the convergence of the contour dynamics method with Euler's explicit method in time. Namely, at terminal time T, the evolving disturbance is going to zero as initial small disturbance goes to zero. The numerical experiment on the stability of contour dynamics has been given in [5,6].     

Vortex dynamics in the presence of excess energy for the Landau–Lifshitz–Gilbert equation

We study the Landau–Lifshitz–Gilbert equation for the dynamics of a magnetic vortex system. We present a PDE-based method for proving vortex dynamics that does not rely on strong well-preparedness of the initial data and allows for instantaneous changes in the strength of the gyrovector force due to bubbling events. The main tools are estimates of the Hodge decomposition of the supercurrent and an analysis of the defect measure of weak convergence of the stress energy tensor. Ginzburg–Landau equations with mixed dynamics in the presence of excess energy are also discussed.     

Vortex design problem

In this investigation we propose a computational approach for the solution of optimal control problems for vortex systems with compactly supported vorticity. The problem is formulated as a PDE-constrained optimization in which the solutions are found using a gradient-based descent method. Recognizing such Euler flows as free-boundary problems, the proposed approach relies on shape differentiation combined with adjoint analysis to determine cost functional gradients. In explicit tracking of interfaces (vortex boundaries) this method offers an alternative to grid-based techniques, such as the level-set methods, and represents a natural optimization formulation for vortex problems computed using the contour dynamics technique. We develop and validate this approach using the design of 2D equilibrium Euler flows with finite-area vortices as a model problem. It is also discussed how the proposed methodology can be applied to Euler flows featuring other vorticity distributions, such as vortex sheets, and to time-dependent phenomena.     

On the hydrodynamic limit of Ginzburg‐Landau wave vortices

In this paper, we study the hydrodynamic limit of the finite Ginzburg‐Landau wave vortices, which was established in [16]. Unlike the classical vortex method for incompressible Euler equations, we prove here that the densities approximated by the vortex blob method associated with the Ginzburg‐Landau wave vortices tend to the solutions of the pressure‐less compressible Euler‐Poisson equations. The convergence of such approximation is proved before the formation of singularities in the limit system as the blob sizes and the grid sizes tend to zero in appropriate rates. © 2002 John Wiley & Sons, Inc.     

Optimal control with connected initial and terminal conditions

An optimal control problem with linear dynamics is considered on a fixed time interval. The ends of the interval correspond to terminal spaces, and a finite-dimensional optimization problem is formulated on the Cartesian product of these spaces. Two components of the solution of this problem define the initial and terminal conditions for the controlled dynamics. The dynamics in the optimal control problem is treated as an equality constraint. The controls are assumed to be bounded in the norm of L₂. A saddle-point method is proposed to solve the problem. The method is based on finding saddle points of the Lagrangian. The weak convergence of the method in controls and its strong convergence in state trajectories, dual trajectories, and terminal variables are proved.     

Vortex Filament Method

In this paper, the convergence of the vortex filament methodfor three-dimensional incompressible and inviscid fluid flowis proved. Properties of consistency and stability are analysed.The foundation for studying these properties is based on thecubature formulae with lines as well as on the specific useof the vector measure that transports the vorticity by the flow,preserving the filament structure of the solution of the problem.In this way, the method takes into account the stretching termimplicitly.     

CONVERGENCE OF VORTEX METHODS FOR 3-D EULER EQUATIONS

1. IntroductionThe convergence problem of vortex methods for the Euler equations has been studied by many authors. Hald and Delprete proved the convergence for two--dimensionalinitial value problems [3]. Three-dimensional initial value problems were studied byBeale and Majda [2] and Beale [1]. Ying [4] and Ying and Zhang [sl, [61 provedthe convergence of vortex methods for two--dimensional initial-boundary value problems of the Euler equations. Ying [7] proved the convergence of vortex met…     

Dynamics and self-propulsion of a spherical body shedding coaxial vortex rings in an ideal fluid

We describe a model for the dynamic interaction of a sphere with uniform density and a system of coaxial circular vortex rings in an ideal fluid of equal density. At regular intervals in time, a constraint is imposed that requires the velocity of the fluid relative to the sphere to have no component transverse to a particular circular contour on the sphere. In order to enforce this constraint, new vortex rings are introduced in a manner that conserves the total momentum in the system. This models the shedding of rings from a sharp physical ridge on the sphere coincident with the circular contour. If the position of the contour is fixed on the sphere, vortex shedding is a source of drag. If the position of the contour varies periodically, propulsive rings may be shed in a manner that mimics the locomotion of certain jellyfish. We present simulations representing both cases.     

Existence and numerical modelling of vortex rings with elliptic boundaries

We revisit in this paper the theory of axisymmetric vortex rings in an ideal fluid. The boundary separating the vortex ring from the external (potential) flow is assumed of elliptic shape. For a given distribution of vorticity in the vortex core, we theoretically put into evidence the critical parameter for the existence of non-trivial solutions, thus confirming the numerical observation of Durst et al. [ZAMP 32 (1981) 156]. A sharp estimation of the critical threshold is analytically derived. Theoretical predictions are confirmed by numerical simulations using finite elements. A new numerical algorithm is presented and shown to display better performances compared to previous published algorithms using finite differences. The convergence of the iterative algorithm is proved using the theory of elliptic partial differential equations with discontinuous nonlinearities.     

A vortex method for computing two-dimensional inviscid incompressible flows

A vortex method is suggested for computing two-dimensional inviscid incompressible flows in a closed domain with a possible flow through it. An algorithm for searching for stable steady vortex configurations is described. The method developed is used to study the dynamics of the Chaplygin-Lamb dipole in a rectangular channel in various flow regimes.     

一类非线性积分微分方程解及其构造

本文讨论了核反应堆动力学中一类具缓发中子的非线性积分微分方程．在一定的条件下证明了该方程在p空间中Mild解、强解、局部解的存在唯一性，以及用迭代法求解的合理性，并估计了收敛速度．     

Low-energy vortex dynamics in the self-dual Chern–Simons–Higgs model

The relativistic Chern–Simons–Higgs theory finds application in anyonic superconductivity and contains topological vortices whose dynamics are poorly understood. The gauge fields are defined by a set of nonlinear constraint equations that can be accurately solved with effective Green’s functions, spectral methods, and a discretization scheme using lattice gauge techniques. Simulations show that low-energy two-vortex interactions are elastic with final scattering angles sensitive to vortex velocity; furthermore, vortex pairs form rotating breather states for certain impact parameters. In this study, a function that reproduces scattering angles in the adiabatic limit for nontangential collisions is presented. Simulation results are discussed in the context of analytical methods that extract vortex dynamics from low-energy effective Lagrangians, and a numerical method to calculate the effective Lagrangian is suggested. The numerical techniques used can be applied to the study of other Chern–Simon theories.     

ON LINEARIZED FINITE DIFFERENCE SIMULATION FOR THE MODEL OF NUCLEAR REACTOR DYNAMICS

1 hoeductIOuThe dynamics models of one--dimensional continuous medium nuclear reactor are the foelowing initial--boundary value problem of the formsubject to the innal conditionsand the boundary conditionsIn (1. 1), x denotes position along the reactor, which is regarded as a rod of length L, t denotes the time, u(t) the logarithm of the loud reactor POwer, v(x,t) the deviation of the temperature from equilibrium, a(x) the ratio of the temperature coefficient of reactivity to theynean life of…     

CHEBYSHEV PSEUDOSPECTRAL-HYBRID FINITE ELEMENT METHOD FOR THREE-DIMENSIONAL VORTICITY EQUATION

In this paper,Chebyshev pseudospectral-finite element schemes are proposed for solving three dimensional vorticity equation.Some approximation results in nonisotropic Sobolev spaces are given.The generalized stability and the convergence are proved strictly.The numerical results show the advantages of this method.The technique in this paper is also applicable to other three-dimensional nonlinear problems in fluid dynamics.     

Generalized Contour Dynamics: A Review

Contour dynamics is a computational technique to solve for the motion of vortices in incompressible inviscid flow. It is a Lagrangian technique in which the motion of contours is followed, and the velocity field moving the contours can be computed as integrals along the contours. Its best-known examples are in two dimensions, for which the vorticity between contours is taken to be constant and the vortices are vortex patches, and in axisymmetric flow for which the vorticity varies linearly with distance from the axis of symmetry. This review discusses generalizations that incorporate additional physics, in particular, buoyancy effects and magnetic fields, that take specific forms inside the vortices and preserve the contour dynamics structure. The extra physics can lead to time-dependent vortex sheets on the boundaries, whose evolution must be computed as part of the problem. The non-Boussinesq case, in which density differences can be important, leads to a coupled system for the evolution of both mean interfacial velocity and vortex sheet strength. Helical geometry is also discussed, in which two quantities are materially conserved and whose evolution governs the flow.     

Lift maximization for a smooth contour placed in flow with a vortex

For a smooth contour placed in an inviscid incompressible potential flow with a vortex, a contour shape and a vortex location are determined for which the lift coefficient is maximal.     

Splitting schemes for the ocean dynamics equations

A splitting scheme in physical processes is proposed for a system of large-scale ocean dynamics equations. The convergence to an exact solution is proved for this scheme.     

Almost optimal convergence of the point vortex method for vortex sheets using numerical filtering

Standard numerical methods for the Birkhoff-Rott equation for a vortex sheet are unstable due to the amplification of roundoff error by the Kelvin-Helmholtz instability. A nonlinear filtering method was used by Krasny to eliminate this spurious growth of round-off error and accurately compute the Birkhoff-Rott solution essentially up to the time it becomes singular. In this paper convergence is proved for the discretized Birkhoff-Rott equation with Krasny filtering and simulated roundoff error. The convergence is proved for a time almost up to the singularity time of the continuous solution. The proof is in an analytic function class and uses a discrete form of the abstract Cauchy-Kowalewski theorem. In order for the proof to work almost up to the singularity time, the linear and nonlinear parts of the equation, as well as the effects of Krasny filtering, are precisely estimated. The technique of proof applies directly to other ill-posed problems such as Rayleigh-Taylor unstable interfaces in incompressible, inviscid, and irrotational fluids, as well as to Saffman-Taylor unstable interfaces in Hele-Shaw cells.