FOURIER－CHEBYSHEV SPECTRAL METHOD FOR SOLVING THREE－DIMENSIONAL VORTICITY EQUATION

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CHEBYSHEV SPECTRAL-FINITE ELEMENTMETHOD FOR THREE-DIMENSIONALVORTICITY EQUATION

1.TheSchemesInthispaper,weconsidercombinedChebyshevspectraLfiniteelementmethodforthreedimensionalunsteadyvorticityequation.LetQbeaconvexpolygoninRZandIbetheinterval(--1,1).x~(xl,x2)andfi={(x,y)/xEQ,yEI}.Theboundaryoffiisdenotedbyoff.Denotethevorticityvectorandstreamvectorby(andoprespectively.Theircomponentsaref(q)andop(q),q=1,2,3.Letu>0bethekineticviscosity.fi,fZandfoaregivenvectors.Thethree-dimensionalvorticityequationisAssumethattheboundaryisafixednon-slipwallandsoop=oonafl.FOrsimpli…     

Error estimation of spectral method for solving three-dimensional vorticity equation

Much work has been done for spectral scheme of P.D.E. (see [1]). Recently the author proposed a technique to prove the strict error estimation of spectral scheme for non-linear problems such as K.D.V.-Burgers' equation, two-dimensional vorticity equation and so on ([2]–[4]). In this paper we generalize this technique into three-dimensional vorticity equation. Under some conditions these error estimations imply convergence. The more smooth the solution of P.D.E., the more accurate the approximate solution.The author is     

Explicit solutions of the viscous model vorticity equation

Explicit solutions are found for the viscous version of the model vorticity equation recently proposed by P. Constantin, P. D. Lax, and A. Majda: where H(w) is the Hilbert transform of w, and v is a positive constant. Various properties of these solutions, including the fact that they blow up after a finite time, are discussed.     

An approximation method for the cauchy problem to the three-dimensional equation of vorticity transport

The vorticity problem (V₀) is shown to have (at least) locally in time a unique classical solution. For numerical purposes global solvability is desired. So by suitable operations we proceed to a family of modified vorticity problems (V_?), ? > 0, possessing a unique classical solution globally in time. For (V_?) a constructive approximation method is introduced. This procedure yields a sequence (ω) of approximate vorticity fields, converging to the global solution of (V_?) and to the local solution of (V₀).     

Derivation of the hydrodynamical equation for one-dimensional Ginzburg-Landau model

Summary The hydrodynamical behavior of one-dimensional scalar Ginzburg-Landau model with conservation law is investigated. The dynamics of the system is given by solving a stochastic partial differential equation. Under appropriate space-time scaling, a deterministic limit is obtained and the limit is described by a certain nonlinear diffusion equation.Dedicated to Professor Takeyuki Hida on his 60th birthday     

A variational difference scheme for the one-dimensional diffusion equation

In this paper we construct a homogeneous variational difference scheme for the diffusion equation assuming its coefficients to be bounded and measurable; the order of convergence of the scheme is O(h²). We consider the boundary value problem (1) $$\frac{d}{{dx}}\left( {K(x)\frac{{du}}{{dx}}} \right) - g(x)u = - \frac{{dF}}{{dx}},0< x< X$$ subject to the boundary conditions (2) $$u(0) = a,u(X) = b$$ .     

Fourier-Chebyshev pseudospectral method for two-dimensional vorticity equation

Summary A Fourier-Chebyshev pseudospectral scheme is proposed for two-dimensional unsteady vorticity equation. The generalized stability and convergence are proved strictly. The numerical results are presented.     

A moment problem for one-dimensional nonlinear pseudoparabolic equation

We are concerned with a moment problem for a nonlinear pseudoparabolic equation with one space dimension on an interval. The boundary conditions are imposed in terms of the zero-order moment and the first-order moment. Based on an elliptic estimate and an iteration method we established the well-posedness of solutions in the usual Sobolev space. We are able to get regularity of the solution so that both solution and its derivative with respect to the time variable belong to the same Sobolev space with respect to the space variable. This feature is different from problems with parabolic equations, where the regularity order of solution is higher than that of the time derivative with respect to the space variable. Previous results reflected only this parabolic nature for the pseudoparabolic equation.     

On blowups of vorticity for the homogeneous Euler equation

Blowups of vorticity for the three- and two-dimensional homogeneous Euler equations are studied. Two regimes of approaching a blow-up point, respectively, with variable or fixed time are analyzed. It is shown that in the n-dimensional ( $n=2,3$) generic case the blowups of degrees $1,\cdots ,n$ at the variable time regime and of degrees $1/2,\cdots ,(n+1)/(n+2)$ at the fixed time regime may exist. Particular situations when the vorticity blows while the direction of the vorticity vector is concentrated in one or two directions are realizable.     

A matrix Bernoulli equation in the adjoint matrix representation of simple three-dimensional Lie algebras

In this paper we give sufficient conditions for solvability by quadratures of a matrix Bernoulli equation whose parameters are defined in the adjointmatrix representation of simple threedimensional Lie algebras over a field of real numbers.     

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.     

Functional equation for the crossover in the model of one-dimensional Weierstrass random walks

We consider the problem of one-dimensional symmetric diffusion in the framework of Markov random walks of the Weierstrass type using two-parameter scaling for the transition probability. We construct a solution for the characteristic Lyapunov function as a sum of regular (homogeneous) and singular (nonhomogeneous) solutions and find the conditions for the crossover from normal to anomalous diffusion.     

Test of a new bosonization algorithm for a simple one-dimensional model

Different aspects of a new bosonization algorithm are tested by numerical simulations of a simple one-dimensional model. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 1, pp. 58–67, October, 1997.     

Galerkin approximations for the one-dimensional vlasov-poisson equation

By taking Fourier series in the space variable, and truncating, we construct a Galerkin scheme for approximate solutions to the one dimensional Vlasov-Poisson equation. Existence is proved for the approximate solutions and they are shown to converge to an exact solution in a weak sense.     

A new high accuracy locally one-dimensional scheme for the wave equation

In this paper, a new locally one-dimensional (LOD) scheme with error of O(Δt⁴+h⁴) for the two-dimensional wave equation is presented. The new scheme is four layer in time and three layer in space. One main advantage of the new method is that only tridiagonal systems of linear algebraic equations have to be solved at each time step. The stability and dispersion analysis of the new scheme are given. The computations of the initial and boundary conditions for the two intermediate time layers are explicitly constructed, which makes the scheme suitable for performing practical simulation in wave propagation modeling. Furthermore, a comparison of our new scheme and the traditional finite difference scheme is given, which shows the superiority of our new method.     

A well-balanced and asymptotic-preserving scheme for the one-dimensional linear Dirac equation

The numerical approximation of one-dimensional relativistic Dirac wave equations is considered within the recent framework consisting in deriving local scattering matrices at each interface of the uniform Cartesian computational grid. For a Courant number equal to unity, it is rigorously shown that such a discretization preserves exactly the \(L^2\) norm despite being explicit in time. This construction is well-suited for particles for which the reference velocity is of the order of \(c\), the speed of light. Moreover, when \(c\) diverges, that is to say, for slow particles (the characteristic scale of the motion is non-relativistic), Dirac equations are naturally written so as to let a “diffusive limit” emerge numerically, like for discrete 2-velocity kinetic models. It is shown that an asymptotic-preserving scheme can be deduced from the aforementioned well-balanced one, with the following properties: it yields unconditionally a classical Schrödinger equation for free particles, but it handles the more intricate case with an external potential only conditionally (the grid should be such that \(c \Delta x\rightarrow 0\)). Such a stringent restriction on the computational grid can be circumvented easily in order to derive a seemingly original Schrödinger scheme still containing tiny relativistic features. Numerical tests (on both linear and nonlinear equations) are displayed.     

Fast numerical method for the solution of the quasigeostrophic vorticity equation

Particle methods are typically O(N²), where N is the number of computational elements. We present an O(N) particle method for the equations for the conservation of potential vorticity. This method is based on the idea of grouping the particles. The necessary expansions and truncation errors are given. The accuracy and speed of the method are presented for both scalar and vector machines. © 1993 John Wiley & Sons, Inc.