COMPACT FINITE DIFFERENCE-FOURIER SPECTRAL METHOD FOR THREE-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

A new compact finite difference-Fourier spectral hybrid method for solving the three dimensional incompressible Navier-Stokes equations is developed in the present paper. The fifth-order upwind compact finite difference schemes for the nonlinear convection terms in the physical space, and the sixth-order center compact schemes for the derivatives in spectral space are described, respectively. The fourth-order compact schemes in a single nine-point cell for solving the Helmholtz equations satisfied by the velocities and pressure in spectral space is derived and its preconditioned conjugate gradient iteration method is studied. The treatment of pressure boundary conditions and the three dimensional non-reflecting outflow boundary conditions are presented. Application to the vortex dislocation evolution in a three dimensional wake is also reported.

Optimal error estimates for Fourier spectral approximation of the generalized KdV equation

A Fourier spectral method for the generalized Korteweg-de Vries equation with periodic boundary conditions is analyzed, and a corresponding optimal error estimate in L^2-norm is obtained. It improves the result presented by Maday and Quarteroni. A modified Fourier pseudospectral method is also presented, with the same convergence properties as the Fourier spectral method.

THE LARGE TIME ERROR ESTIMATES OF FOURIER SPECTRAL METHOD FOR GENERALIZED BENJAMIN-BONA-MAHONY EQUATIONS

In this paper, a spectral method to analyze the generalized Benjamin Bona Mahony equations is used. The existence and uniqueness of global smooth solution of these equations are proved. The large time error estimation between the spectral approximate solution and the exact solution is obtained.

Fourier time spectral method for subsonic and transonic flows

The time accuracy of the exponentially accurate Fourier time spectral method（TSM） is examined and compared with a conventional 2nd-order backward difference formula（BDF） method for periodic unsteady flows. In particular, detailed error analysis based on numerical computations is performed on the accuracy of resolving the local pressure coefficient and global integrated force coefficients for smooth subsonic and non-smooth transonic flows with moving shock waves on a pitching airfoil. For smooth subsonic flows, the Fourier TSM method offers a significant accuracy advantage over the BDF method for the prediction of both the local pressure coefficient and integrated force coefficients. For transonic flows where the motion of the discontinuous shock wave contributes significant higherorder harmonic contents to the local pressure fluctuations,a sufficient number of modes must be included before the Fourier TSM provides an advantage over the BDF method.The Fourier TSM, however, still offers better accuracy than the BDF method for integrated force coefficients even for transonic flows. A problem of non-symmetric solutions for symmetric periodic flows due to the use of odd numbers of intervals is uncovered and analyzed. A frequency-searching method is proposed for problems where the frequency is not known a priori. The method is tested on the vortex shedding problem of the flow over a circular cylinder.

Numerical Simulation of Space Fractional Order Schnakenberg Model

A numerical solution of a fractional-order reaction-diffusion model is discussed.With the development of fractional-order differential equations,Schnakenberg model becomes more and more important.However,there are few researches on numerical simulation of Schnakenberg model with spatial fractional order.It is also important to find a simple and effective numerical method.In this paper,the Schnakenberg model is numerically simulated by Fourier spectral method.The Fourier transform is applied to transforming the partial differential equation into ordinary differential equation in space,and the fourth order Runge-Kutta method is used to solve the ordinary differential equation to obtain the numerical solution from the perspective of time.Simulation results show the effectiveness of the proposed method.

Heat Convection Between Two Confocal Elliptic Tubes Placed at Different Orientations

In this paper,transient and steady natural convection heat transfer in an elliptical annulus has been investigated.The annulus occupies the space between two horizontal concentric tubes of elliptic cross-section.The resulting velocity and thermal fields are predicted at different annulus orientations assuming isothermal surfaces.The full governing equations of mass,momentum and energy are solved numerically using the Fourier Spectral method.The heat convection process between the two tubes depends on Rayleigh number,Prandtl number,angle of inclination of tube axes and the geometry and dimensions of both tubes.The Prandtl number and inner tube axis ratio are fixed at 0.7 and 0.5,respectively.The problem is solved for the two Rayleigh numbers of 104 and 105 considering a ratio between the two major axes up to 3 while the angle of orientation of the minor axes varies from 0 to 90◦.The results for local and average Nusselt numbers are obtained and discussed together with the details of both flow and thermal fields.For isothermal heating conditions,the study has shown an optimum value for major axes ratio that minimizes the rate of heat transfer between the two tubes.Another important aspect of this paper is to prove the successful use of the Fourier Spectral Method in solving confined flow and heat convection problems.

Novel Multi-Symplectic Integrators for Nonlinear Fourth-Order Schrodinger Equation with Trapped Term

The multi-symplectic Runge-Kutta (MSRK) methods and multi-symplecticFourier spectral (MSFS) methods will be employed to solve the fourth-orderSchrodinger equations with trapped term. Using the idea of split-step numericalmethod and the MSRK methods, we devise a new kind of multi-symplectic integrators, which is called split-step multi-symplectic (SSMS) methods. The numerical experiments show that the proposed SSMS methods are more efficient than the conventionalmulti-symplectic integrators with respect to the the numerical accuracy and conservation perserving properties.

Analysis of Mathematics and Numerical Pattern Formation in Superdiffusive Fractional Multicomponent System

In this work,we examine the mathematical analysis and numerical simulation of pattern formation in a subdiffusive multicomponents fractional-reactiondiffusion system that models the spatial interrelationship between two preys and predator species.The major result is centered on the analysis of the system for linear stability.Analysis of the main model reflects that the dynamical system is locally and globally asymptotically stable.We propose some useful theorems based on the existence and permanence of the species to validate our theoretical findings.Reliable and efficient methods in space and time are formulated to handle any space fractional reaction-diffusion system.We numerically present the complexity of the dynamics that are theoretically discussed.The simulation results in one,two and three dimensions show some amazing scenarios.