Homogenization and convergence of the vortex method for 2-D euler equations with oscillatory vorticity fields

The 2-D incompressible Euler equations with oscillatory vorticity fields are studied. A homogenization result for 2-D Euler equations in velocity-vorticity formulation is obtained and weak continuity of the equations is proved. Convergence of the vortex method is analyzed in the case when the continuous vorticity is not well resolved by the computational particles. Numerical results are given. Comparisons are made with the corresponding finite difference approximation.     

Convergence of vortex methods for weak solutions to the 2-D euler equations with vortex sheet data

We prove the convergence of vortex blob methods to classical weak solutions for the two-dimensional incompressible Euler equations with initial data satisfying the conditions that the vorticity is a finite Radon measure of distinguished sign and the kinetic energy is locally bounded. This includes the important example of vortex sheets. The result is valid as long as the computational grid size h does not exceed the smoothing blob size ε, i.e., h/ε ≦ C.. ©1995 John Wiley & Sons, Inc.     

Error indicators for the Navier-Stokes equations in stream function and vorticity formulation

This paper deals with a posteriori estimates for the finite element solution of the Stokes problem in stream function and vorticity formulation. For two different discretizations, we propose error indicators and we prove estimates in order to compare them with the local error. In a second step, these results are extended to the Navier-Stokes equations. Received March 25, 1996 / Revised version received April 7, 1997     

Numerical solution of the time-dependent incompressible Navier-Stokes equations in the stream function and vorticity formulation

In this work we study the time-dependent incompressible Navier-Stokes problem. We introduce a suitable technique based on the splitting of the vorticity into two components. Then we discretize in space the resulting uncoupled system by means of continuous Lagrange finite elements. This is achieved by first performing the semi-discretization in time of these equations by a classical characteristics method for the advective term. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of finite difference schemes for unsteady Navier-Stokes equations in vorticity formulation

Summary. In this paper, we provide stability and convergence analysis for a class of finite difference schemes for unsteady incompressible Navier-Stokes equations in vorticity-stream function formulation. The no-slip boundary condition for the velocity is converted into local vorticity boundary conditions. Thom's formula, Wilkes' formula, or other local formulas in the earlier literature can be used in the second order method; while high order formulas, such as Briley's formula, can be used in the fourth order compact difference scheme proposed by E and Liu. The stability analysis of these long-stencil formulas cannot be directly derived from straightforward manipulations since more than one interior point is involved in the formula. The main idea of the stability analysis is to control local terms by global quantities via discrete elliptic regularity for stream function. We choose to analyze the second order scheme with Wilkes' formula in detail. In this case, we can avoid the complicated technique necessitated by the Strang-type high order expansions. As a consequence, our analysis results in almost optimal regularity assumption for the exact solution. The above methodology is very general. We also give a detailed analysis for the fourth order scheme using a 1-D Stokes model. Received December 10, 1999 / Revised version received November 5, 2000 / Published online August 17, 2001     

Spectral discretization of the vorticity, velocity and pressure formulation of the Navier–Stokes equations

We consider the Navier–Stokes equations in a two- or three-dimensional domain provided with non standard boundary conditions which involve the normal component of the velocity and the tangential components of the vorticity. We write a variational formulation of this problem with three independent unknowns, the vorticity, the velocity and the pressure, and prove the existence of a solution for this problem. Next we propose a discretization by spectral methods which relies on this formulation. In the two-dimensional case, we prove quasi-optimal error estimates for the three unknowns. We conclude with some numerical experiments.

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Résumé Nous considérons les équations de Navier–Stokes dans un domaine biou tri-dimensionnel, munies de conditions aux limites non usuelles portant sur la composante normale de la vitesse et la ou les composantes tangentielles du tourbillon. Nous écrivons une formulation variationnelle de ce problème qui comporte trois inconnues indépendantes: le tourbillon, la vitesse et la pression. Nous prouvons que ce problème admet au moins une solution. Nous proposons une discrétisation par méthodes spectrales construite à partir de cette formulation. Dans le cas bidimensionnel, nous établissons des majorations quasi-optimales de l'erreur pour les trois inconnues. Nous concluons par quelques expériences numériques.

     

Convergence of the point vortex method for the 2-D euler equations

We prove consistency, stability and convergence of the point vortex approximation to the 2-D incompressible Euler equations with smooth solutions. We first show that the discretization error is second-order accurate. Then we show that the method is stable in l_p norm. Consequently the method converges in l_p norm for all time. The convergence is also illustrated by a numerical experiment.     

A lagrangian finite element method for the 2-D euler equations

A grid free Lagrangian finite element method is introduced for the 2-D incompressible Euler equations. The method is derived based on the observation that the product of the Biot-Savart kernel and a polynomial can be integrated analytically over any triangle. This enables us to obtain a numerically stable Lagrangian method without using numerical smoothing. Moreover, we show that the method converges uniformly with second-order accuracy. Actually, we establish a l^∞ stability result which applies to kernels that are more singular than the Biot-Savart kernel, as long as the kernel is L integrable. Another useful result is that we prove convergence of our method when using local regridding, which allows the method to run for longer time even with a fixed mesh. The second-order convergence is also illustrated by our numerical experiments.     

Backward stochastic differential equations associated with the vorticity equations

In this paper we derive a non-linear version of the Feynman–Kac formula for the solutions of the vorticity equation in dimension 2 with space periodic boundary conditions. We prove the existence (global in time) and uniqueness for a stochastic terminal value problem associated with the vorticity equation in dimension 2. A particular class of terminal values provide, via these probabilistic methods, solutions for the vorticity equation.     

A weak formulation of boundary integral equations for time dependent parabolic problems

A weak formulation for ‘direct’ boundary methods for time dependent parabolic problems, deduced from distribution theory, is presented. The present approach seems particularly valuable when dealing with problems with non-integrable singularities and solutions with an exponential growth. Numerical examples are also reported for plane diffusion.     

Note on the compressible euler equations with zero temperature

This note presents the behavior of solution of the compressible Euler equations as the temperature drops to zero by the simple riemann problem.     

A block preconditioning technique for the streamfunction‐vorticity formulation of the Navier‐Stokes equations

Iterative methods of Krylov‐subspace type can be very effective solvers for matrix systems resulting from partial differential equations if appropriate preconditioning is employed. We describe and test block preconditioners based on a Schur complement approximation which uses a multigrid method for finite element approximations of the linearized incompressible Navier‐Stokes equations in streamfunction and vorticity formulation. By using a Picard iteration, we use this technology to solve fully nonlinear Navier‐Stokes problems. The solvers which result scale very well with problem parameters. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Meshless schemes for unsteady Navier–Stokes equations in vorticity formulation using radial basis functions

In this paper, we provide a new scheme for unsteady incompressible flows in vorticity-stream function formulation. Combined with the radial basis functions method, it is an efficient meshless method. Optimal accuracy can be achieved using this method. The efficiency and accuracy are demonstrated by numerical examples.     

The lifespan for 3-D spherically symmetric compressible euler equations

In this paper, we study the lifespan of solutions for three dimensional compressible Euler equations with spherically symmetric initial data that is a small perturbation of amplitude ε from a constant state. From our result, the classical solutions have to blow up in finite time in spite of any small ε. Project supported by the Tianyuan Foundation of China, the National Natural Science Foundation of China and Lab of Mathematics for Nonlinear Problems. Fudan University     

The fourier pseudospectral method for three-dimensional vorticity equations

This paper develops the Fourier Pseudospectral Method to solve three-dimensional vorticity equations. The generalized stability is proved, from which the convergence follows with some assumptions.