SPECTRAL METHOD IN TIME FOR KdV EQUATIONS

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A return toward equilibrium in a 2D Rayleigh–Taylor instability for compressible fluids with a multidomain adaptive Chebyshev method

We numerically simulate a single-mode Rayleigh–Taylor instability between compressible miscible fluids with a highly accurate self-adaptive pseudospectral Chebyshev multidomain method in two two-dimensional boxes at small aspect ratios. The simulations are started from rest and pursued until the return toward mechanical equilibrium of the mixing. Four regimes—linear and weakly nonlinear, nonlinear steady bubble rise, return toward equilibrium, and finally a system of acoustic waves—can be identified. We show that this one-dimensional system of stationary acoustic waves is damped by the physical viscosity. This provides a reference solution.      

A PSEUDOSPECTRAL σ -TRANSFORMATION MODEL OF 2-D NONLINEAR WAVES

A pseudospectral matrix-element method is proposed for the analysis of 2-D nonlinear time-domain free-surface flow problems. The Chebyshev expansion technique established by Ku & Hatziavramidis has been used to discretize the σ-transformed governing equations including nonlinear boundary conditions. Simulations of non overturning transient waves in fixed and base-excited tanks are presented. The results are compared with first-and second-order analytical solutions for sloshing and standing waves, respectively. Excellent agreement is achieved at low values of wave steepness, with the high accuracy due to the close coupling between points. As the wave steepness increases, the influence of higher-order nonlinear components becomes significant, and is modelled by the present scheme. The solution is extremely stable, with the σ-transformation exactly fitting the free-surface boundary, unlike other schemes which have to use free-surface smoothing.     

Pseudospectral analysis of the interfacial stability of free jets with different velocity profiles

A double fluid model for a liquid jet surrounded by a coaxial gas stream was constructed. The interfacial stability of the model was studied by Chebyshev pseudospectral method for different basic velocity profiles. The physical variables were mapped into computational space using a nonlinear coordinates transformation. The general eigenvalues of the dispersion relation obtained are solved by QZ method, and the basic characteristics and their dependence on the flow parameters are analyzed.     

A high-accuracy temporal–spatial pseudospectral method for time-periodic unsteady fluid flow and heat transfer problems

A temporal–spatial pseudospectral (TSP) method is proposed for the high-accuracy solutions of time-periodic unsteady fluid flow and heat transfer problems. In this method, both the spatial and temporal derivative terms in the governing equations are computed by pseudospectral method. The spatial derivatives are computed through Chebyshev and Lagrange polynomials while the time derivatives are computed by Fourier series. The TSP method is capable of directly finding out the periodic state solutions without the necessity to resolve the initial transient state solutions, hence holds high computational efficiency and high numerical accuracy properties for the time-periodic problems. This method is validated by three 2D benchmark problems: the time-periodic incompressible flow with exact solutions; the natural convection in enclosure with time-periodic temperature on one sidewall, and on both sidewalls. The TSP results fit well the exact solutions or the benchmark solutions and the TSP accuracy is much higher than the time marching spatial pseudospectral accuracy. Some time-dependent fluid flow and heat transfer characteristic parameters are analysed. The proposed TSP method could be further extended to more complex time-periodic unsteady fluid flow and heat transfer problems where high-accuracy results are required.     

Generalized Euler–Lagrange equation for nonsmooth calculus of variations

In this paper, we first introduce a novel generalized derivative and obtain the generalized first-order Taylor expansion of the nonsmooth functions. Then we derive the generalized Euler–Lagrange equation for the nonsmooth calculus of variations and solve this equation by using Chebyshev pseudospectral method, approximately. Finally, the optimal solutions of some problems in the nonsmooth calculus of variations are approximated.     

Linear and Nonlinear Stability Analysis of a Horton–Rogers–Lapwood Problem with an Internal Heat Source and Brinkman Effects

The effect of internal heat source on convection in a layer of fluid in a porous medium was analyzed using linear and nonlinear analysis, and boundaries are assumed to be stress-free and isothermal. Normal mode technique is used for linear analysis, and energy method is used for nonlinear stability analysis. The presence of heat generation leads to the possibility of the existence of a subcritical instability. Effects of increase of Darcy–Brinkman number and internal heat parameter on critical Rayleigh numbers were analyzed numerically using Chebyshev pseudospectral method.     

Modified Laguerre spectral and pseudospectral methods for nonlinear partial differential equations in multiple dimensions

The Laguerre spectral and pseudospectral methods are investigated for multidimensional nonlinear partial differential equations. Some results on the modified Laguerre orthogonal approximation and interpolation are established, which play important roles in the related numerical methods for unbounded domains. As an example, the modified Laguerre spectral and pseudospectral methods are proposed for two-dimensional Logistic equation. The stability and convergence of the suggested schemes are proved. Numerical results demonstrate the high accuracy of these approaches.     

含有界随机参数的双势井Duffing-Van der pol系统的对称破裂分岔

讨论谐和激励作用下含有界随机参数的双势井Duffing-Van der pol系统的对称破裂分岔现象。首先用Chebyshev多项式逼近法将随机系统化成与其等价的确定性系统,然后通过等价确定性系统来探索随机Duffing-Van der pol系统的对称破裂分岔现象。数值模拟显示随机Duffing-Van der pol系统与确定性均值参数系统有着类似的对称破裂分岔行为,文中的主要数值结果表明Chebyshev多项式逼近法是研究非线性随机参数系统动力学问题的一种有效方法。     

A Chebyshev collocation method for the Navier–Stokes equations with application to double-diffusive convection

A Chebyshev collocation method for solving the unsteady two-dimensional Navier–Stokes equations in vorticity–streamfunction variables is presented and discussed. The discretization in time is obtained through a class of semi-implicit finite difference schemes. Thus at each time cycle the problem reduces to a Stokes-type problem which is solved by means of the influence matrix technique leading to the solution of Helmholtz-type equations with Dirichlet boundary conditions. Theoretical results on the stability of the method are given. Then a matrix diagonalization procedure for solving the algebraic system resulting from the Chebyshev collocation approximation of the Helmholtz equation is developed and its accuracy is tested. Numerical results are given for the Stokes and the Navier–Stokes equations. Finally the method is applied to a double-diffusive convection problem concerning the stability of a fluid stratified by salinity and heated from below.     

Chebyshev collocation method and multidomain decomposition for the incompressible Navier-Stokes equations

The two-dimensional incompressible Navier-Stokes equations in primitive variables have been solved by a pseudospectral Chebyshev method using a semi-implicit fractional step scheme. The latter has been adapted to the particular features of spectral collocation methods to develop the monodomain algorithm. In particular, pressure and velocity collocated on the same nodes are sought in a polynomial space of the same order; the cascade of scalar elliptic problems arising after the spatial collocation is solved using finite difference preconditioning. With the present procedure spurious pressure modes do not pollute the pressure field. As a natural development of the present work a multidomain extent was devised and tested. The original domain is divided into a union of patching sub-rectangles. Each scalar problem obtained after spatial collocation is solved by iterating by subdomains. For steady problems a C¹ solution is recovered at the interfaces upon convergence, ensuring a spectrally accurate solution. A number of test cases have been solved to validate the algorithm in both its single-block and multidomain configurations. The preliminary results achieved indicate that collocation methods in multidomain configurations might become a viable alternative to the spectral element technique for accurate flow prediction.     

Nonlinear convective stability problems of viscoelastic fluids in finite domains

A Chebyshev pseudospectral method is generalized to solve the nonlinear hydrodynamic stability problems of Rayleigh-Bénard convection of viscoelastic fluids in finite domains, which are compatible with the experimental situations, for the range of viscoelastic parameters where the exchange of stabilities is valid. The effects of box aspect ratio, the Deborah number 5 and the dimensionless retardation time ) on the critical Rayleigh number and convection intensity are investigated. The comparison of these results with the experimental data might be used to guide the selection of constitutive equations and to estimate viscoelastic parameter values. The present technique of hydrodynamic stability analysis is quite versatile and can be employed to solve other hydrodynamic stability problems in finite domains.     

Effect of double dispersion on non-Darcy mixed convective flow over vertical surface embedded in porous medium

A numerical study of a non-Darcy mixed convective heat and mass transfer flow over a vertical surface embedded in a dispersion, melting, and thermal radiation is porous medium under the effects of double investigated. The set of governing boundary layer equations and the boundary conditions is transformed into a set of coupled nonlinear ordinary differential equations with the relevant boundary conditions. The transformed equations are solved numerically by using the Chebyshev pseudospectral method. Comparisons of the present results with the existing results in the literature are made, and good agreement is found. Numerical results for the velocity, temperature, concentration profiles, and local Nusselt and Sherwood numbers are discussed for various values of physical parameters.     

A three-dimensional PSME model for rotating flows in an annular cavity

A pseudospectral matrix-element (PSME) numerical model is described for the simulation of rotating flows in a three-dimensional annular cavity. Temporal discretisation is implemented using a second-order semi-implicit scheme. Modified compressibility is invoked to handle the coupling between velocity and pressure while maintaining the incompressibility constraint. The governing continuity and Navier–Stokes momentum equations and boundary conditions are discretised using Chebyshev and Fourier collocation formulae. The model is validated against numerical results from alternative schemes and experimental data on rotating flows in an annular cavity. A base flow regime and instability patterns are observed, in accordance with other previously published investigations. It is demonstrated that the PSME model provides an accurate representation of rotating flows in an annular cavity.     

Wave propagation analysis in anisotropic and inhomogeneous uncracked and cracked structures using pseudospectral finite element method

The work deals with the development of an effective numerical tool in the form of pseudospectral method for wave propagation analysis in anisotropic and inhomogeneous structures. Chebyshev polynomials are used as basis functions and Chebyshev–Gauss–Lobatto points are used as grid points. The formulation is implemented in the same way as conventional finite element method. The element is tested successfully on a variety of problems involving isotropic, orthotropic and functionally graded material (FGM) structures. The formulation is validated by performing static, free vibration and wave propagation analysis. The accuracy of the element in predicting stresses is compared with conventional finite elements. Free vibration analysis is carried out on composite and FGM beams and the computational resources saved in each case are presented. Wave propagation analysis is carried out using the element on anisotropic and inhomogeneous beams and layer structures. Wave propagation in thin double bounded media over long propagating distances is studied. Finally, a study on scattering of waves due to embedded horizontal and vertical cracks is carried out, where the effectiveness of modulated pulse in detecting small cracks in composites and FGMs has been demonstrated.     

Direct spectral domain decomposition method for 2D incompressible Navier-Stokes equations

An efficient direct spectral domain decomposition method is developed coupled with Chebyshev spectral approximation for the solution of 2D, unsteady and incompressible Navier-Stokes equations in complex geometries. In this numerical approach, the spatial domains of interest are decomposed into several non-overlapping rectangular sub-domains. In each sub-domain, an improved projection scheme with second-order accuracy is used to deal with the coupling of velocity and pressure, and the Chebyshev collocation spectral method (CSM) is adopted to execute the spatial discretization. The influence matrix technique is employed to enforce the continuities of both variables and their normal derivatives between the adjacent sub-domains. The imposing of the Neumann boundary conditions to the Poisson equations of pressure and intermediate variable will result in the indeterminate solution. A new strategy of assuming the Dirichlet boundary conditions on interface and using the first-order normal derivatives as transmission conditions to keep the continuities of variables is proposed to overcome this trouble. Three test cases are used to verify the accuracy and efficiency, and the detailed comparison between the numerical results and the available solutions is done. The results indicate that the present method is efficiency, stability, and accuracy.     

Numerical Modeling of the Performance of Lubricated Journal Bearings Using Navier-Stokes Equations

Two-dimensional incompressible Navier-Stokes equations are solved numerically to model the thermohydrodynamic performance of a dynamically loaded journal bearing which is modeled as eccentrically rotating cylinders. The region between those cylinders are occupied by Newtonian lubricants, whose physical properties such as viscosity and thermal conductivity are assumed to be functions of local temperature. A single domain pseudospectral method which combines Fourier expansions and Chebyshev polynomials for spatial discretization is introduced in conjunction with appropriate time marching scheme for the unsteady incompressible Navier-Stokes equations. The selection of these polynomial functions is favorable since both FFT algorithms for Fourier and Chebyshev expansions are easily available. In this numerical model, the journal is dynamically loaded by an external force and set free, so that its center moves in such a way to strike a balance between the applied load and the hydrodynamic forces. The pseudo-spectral scheme is then applied to a few classical problems, such as concentric rotating cylinders and journal bearings with lubricants of constant and varying (temperature dependent) viscosity to establish the validity of the numerical scheme in simulating these problems realistically as well as to gauge the convergence characteristics and relevant numerical issues. The numerical modeling has been found to be reasonably accurate and robust enough to serve as a tool for the study the flow in the region between the journal and the bearing.     

Bézier curves in the modification of boundary integral equations (BIE) for potential boundary-values problems

The paper presents a modification of the classical boundary integral equation method (BIEM) for two-dimensional potential boundary values problems. The proposed modification consists in describing the boundary geometry by means of Bézier curves. As a result of this analytical modification of the BIEM, a new parametric integral equation system (PIES) was obtained. The kernels of these equations include the geometry of the boundary. This new PIES is no longer defined on the boundary, as in the case of the BIEM, but on the straight line for any given domain. The solution of the new PIES does not require a boundary discretization since it can be reduced merely to an approximation of boundary functions. To solve this PIES a pseudospectral method has been proposed and the results obtained were compared with exact solutions.     

Application of Chebyshev polynomials to the theory of thin bodies

The application of shifted Chebyshev polynomials of the second kind to the construction of the thin-body theory is considered. Some basic and additional recurrence relations for Chebyshev polynomials are given. Arbitrary-order moments are obtained for the first and second derivatives of a tensor field and for some expressions. Several equations of motion expressed in terms of the moments of displacement and rotation vectors are derived for the moment theory; a number of constitutive relations of the zeroth approximation are obtained.     

勒让德伪谱法求解三维刚体摆姿态运动最优控制问题

研究伪谱法求解三维刚体摆姿态运动最优控制问题. 针对三维刚体摆这类含有约束的力学模型,提出了基于勒让德伪谱法的三维刚体摆姿态最优控制方法. 利用插值逼近设计了三维刚体摆姿态运动最优控制算法,得到了三维刚体摆的姿态最优控制轨迹,并结合松弛参数来控制插值点的取值,寻找满足的可行解. 仿真结果表明,基于勒让德伪谱法的最优控制算法使得三维刚体摆能以较小的误差运动到期望的末端姿态,且计算速度快,能够获得精度较高的控制输入量.