Fourier-Chebyshev Pseudospectral Method for Three-Dimensional Vorticity Equation

1.IntroductionInstudyingboundarylayers,flowspastsllddenlyheatedverticalplatesandotherrelatedproblems,we.havetoconsiderbilaterallyperiodicproblems.Thereareseveralwaystosolvethemnumerically.Forinstance,Murdok[1],Macaraeg[2]andBenyuGuo,Yue-shanXiong[31proposedspectral--differenceschemes,whileCanuto,Maday,Quarteroni[41andGuoBen--yu,CaoWeiMing[']developedspectral-finiteelementschemes.Buttheaccuracyofalltheseschemesisstilllimitedduetofinitedifferenceandfiniteelementapproximations,evenifthegenu…     

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 PSEUDOSPECTRAL METHOD FOR VORTICITY EQUATIONS ON SPHERICAL SURFACE

ThisresearchissupportedbytheNationalNaturalScienceFoundationofChina(1990-1992)andtheNaturalScienceFoundationofShanghai(1991-1993).1.IntroductionInfluiddynamics,numericalweatherpredictionandotherengineeringfields,therearelotsofpartialdifferentialequationsdefinedonsphericalsurface[1--3].Ofcourse,finitedifferenceandfiniteelementmethodsareapplicabletotheseproblemsI4].Buttheirconvergenceratesareusuallyrestricted.Ontheotherhand,withthesocalled"infiniteorder"ofconvergence,spectralmethodshavebeen…     

CHEBYSHEV SPECTRAL-FINITE ELEMENT METHOD FOR INCOMPRESSIBLE FLUID FLOW

In this paper, we propose a mixed method for solving two-dimensional unsteady vorticity equations by using Chebyshev spectral-fiuite element approximation. The generalized stability and the optimal rate of convergence are proved. The numerical results show the advantages of such method. The technique in this paper is also useful for other nonlinear problems.     

The Crank–Nicolson finite spectral element method and numerical simulations for 2D non-stationary Navier–Stokes equations

In this paper, we first build a semi-discretized Crank–Nicolson (CN) model about time for the two-dimensional (2D) non-stationary Navier–Stokes equations about vorticity–stream functions and discuss the existence, stability, and convergence of the time semi-discretized CN solutions. And then, we build a fully discretized finite spectral element CN (FSECN) model based on the bilinear trigonometric basic functions on quadrilateral elements for the 2D non-stationary Navier–Stokes equations about the vorticity–stream functions and discuss the existence, stability, and convergence of the FSECN solutions. Finally, we utilize two sets of numerical experiments to check out the correctness of theoretical consequences.     

Remarks on the Regularity to 3-D Ideal Magnetohydrodynamic Equations

In this paper we are interested in the sufficient conditions which guarantee the regularityof solutions of 3-D ideal magnetohydrodynamic equations in the arbitrary time interval [0,T].Fivesufficient conditions are given.Our results are motivated by two main ideas:one is to control theaccumulation of vorticity alone;the other is to generalize the corresponding geometric conditions of3-D Euler equations to 3-D ideal magnetohydrodynamic equations.     

A novel numerical method and its convergence for nonlinear delay Volterra integro-differential equations

In this paper, we suggest a convergent numerical method for solving nonlinear delay Volterra integro-differential equations. First, we convert the problem into a continuous-time optimization problem and then use a shifted pseudospectral method to discrete the problem. Having solved the last problem, we can achieve the pointwise and continuous approximate solutions for the main delay Volterra integro-differential equations. Here, we analyze the convergence of the method and solve some numerical examples to show the efficiency of the method.     

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     

A Pseudospectral Method for Solving Navier-Stokes Equations

In this paper, we propose a new kind of pseudospectral schemes with a restraint operator to solve the periodic problem of Navier-Stokes equations. The generalized stability of the schemes is analysed and convergence is proved. Numerical results are presented also.     

Spectral-finite element method for solving two-dimensional vorticity equations

In this paper, we construct a spectral-finite element scheme for solving semi-periodical two-dimensional vorticity equations. The error between the genuine solution and approximate solution is estimated strictly. The numerical results show the advantages of such a method. The technique used in this paper can be easily generalized to three-dimensional problems.     

An efficient Levenberg–Marquardt method with a new LM parameter for systems of nonlinear equations

A modified Levenberg–Marquardt method for solving singular systems of nonlinear equations was proposed by Fan [J Comput Appl Math. 2003;21;625–636]. Using trust region techniques, the global and quadratic convergence of the method were proved. In this paper, to improve this method, we decide to introduce a new Levenberg–Marquardt parameter while also incorporate a new nonmonotone technique to this method. The global and quadratic convergence of the new method is proved under the local error bound condition. Numerical results show the new algorithm is efficient and promising.     

The Convergence of the Spectral Scheme for Solving Two-Dimensional Vorticity Equations

Much work has been done for the spectral scheme of the P.D.E. The author proposed a technique to prove the strict error estimation of the spectral scheme for the K.D.V.-Burgers equation. In this paper, the technique is generalized to two-dimensional vorticity equations. Under some conditions, the error estimation implies the convergence. The more smooth the solution of the vorticity equations, the more accurate the approximate solution.     

Bernstein Polynomial Collocation Method for Elliptic Boundary Value Problems

We present a summary of recent developments in application of Bernstein polynomials to solution of elliptic boundary value problems with a pseudospectral method. Solution is approximated using Benstein polynomial interpolant defined at points of the extrema of Chebyshev polynomials i.e. the Chebyshev-Gauss-Lobatto (CGL) nodes. This approach brings impovement comparing to the Bernstein interpolation at equidistant nodes we used previously [1]. We show that this approach leads to spectral convergence and accuracy comparable to that of pseudospectral methods with orthogonal polynomials (Chebyshev, Legendre). The algorithm is implemented in open source library bernstein-poly , which is available online. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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…     

Convergence of the point vortex method for the 3-D euler equations

We prove consistency, stability, and convergence of a point vortex approximation to the 3-D incompressible Euler equations with smooth solutions. The 3-D algorithm we consider here is similar to the corresponding 3-D vortex blob algorithm introduced by Beale and Majda; see [3]. We first show that the discretization error is second-order accurate. Then we show that the method is stable in l^p norm for the particle trajectories and in w^?1.p norm for discrete vorticity. Consequently, the method converges up to any time for which the Euler equations have a smooth solution. One immediate application of our convergence result is that the vortex filament method without smoothing also converges.     

Pseudospectral method for quadrilaterals

In this paper, we investigate the pseudospectral method on quadrilaterals. Some results on Legendre-Gauss-type interpolation are established, which play important roles in the pseudospectral method for partial differential equations defined on quadrilaterals. As examples of applications, we propose pseudospectral methods for two model problems and prove their spectral accuracy in space. Numerical results demonstrate the efficiency of the suggested algorithms. The approximation results and techniques developed in this paper are also applicable to other problems defined on quadrilaterals.     

Multi-grid methods for stokes and navier-stokes equations

Summary In the present paper we introduce transforming iterations, an approach to construct smoothers for indefinite systems. This turns out to be a convenient tool to classify several well-known smoothing iterations for Stokes and Navier-Stokes equations and to predict their convergence behaviour, epecially in the case of high Reynolds-numbers. Using this approach, we are able to construct a new smoother for the Navier-Stokes equations, based on incomplete LU-decompositions, yielding a highly effective and robust multi-grid method. Besides some qualitative theoretical convergence results, we give large numerical comparisons and tests for the Stokes as well as for the Navier-Stokes equations. For a general convergence theory we refer to [29].This work was supported in part by Deutsche Forschungsgemeinschaft     

On the Chebyshev spectral continuous time approximation for constant and periodic delay differential equations

In this paper, the approximation technique proposed in Breda et al. (2005) [1] for converting a linear system of constant-coefficient delay differential equations (DDEs) into a system of ordinary differential equations (ODEs) using pseudospectral differencing is extended to linear and nonlinear systems of DDEs with time-periodic coefficients. The Chebyshev spectral continuous time approximation (ChSCTA) technique is used to study the stability of first and second-order constant coefficient DDEs, a delayed system with a cubic nonlinearity and parametric sinusoidal excitation, the delayed Mathieu’s equation, and delayed systems with two fixed delays. In all the examples, the stability and time response obtained from ChSCTA show good agreement with either analytical results, or the results obtained before by other reliable approximation methods. The “spectral accuracy” convergence behavior of Chebyshev spectral collocation shown in Trefethen (2000) [2] which the proposed technique possesses is compared to the convergence properties of finite difference-based continuous time approximation for constant-coefficient DDEs proposed recently in Sun (2009) [3] and Sun and Song (2009) [4].     

Pseudospectral method for quadrilaterals

In this paper, we investigate the pseudospectral method on quadrilaterals. Some results on Legendre–Gauss-type interpolation are established, which play important roles in the pseudospectral method for partial differential equations defined on quadrilaterals. As examples of applications, we propose pseudospectral methods for two model problems and prove their spectral accuracy in space. Numerical results demonstrate the efficiency of the suggested algorithms. The approximation results and techniques developed in this paper are also applicable to other problems defined on quadrilaterals.     

A Spectral Method for a Class of Nonlinear Quasi-Parabolic Equations

1.IntroductionThispaperconsidersthefollowingnonlinearquasi-parabolicequationsofhigherorderwithperiodicboundaryconditions:Here,u(x,t)isavectorfunctionwithdimensionJu(x,t)=(u1(x,t),...9uJ(x,t)),A=(ai,j)ij=,isasymmetricandpositivedefinitematrix,andaijarerealconstants,i.e.(1)takesthefo1lowingform:werehj(u)isafunctionofthevectorultheJacobimatrixofh=(hj)j,issemi-bounded,i'e.thereexistsaconstantbsuchthatASpectralMeth0df0raClass('f\,)11li1l`..1r()llJsiIJaraI)')licI:q11ati(f)lls89F=F(PO,...iPM-l…