Focused Wave Properties Based on A High Order Spectral Method with A Non-Periodic Boundary

In this paper, a numerical model is developed based on the High Order Spectral （HOS） method with a non-periodic boundary. A wave maker boundary condition is introduced to simulate wave generation at the incident boundary in the HOS method. Based on the numerical model, the effects of wave parameters, such as the assumed focused amplitude, the central frequency, the frequency bandwidth, the wave amplitude distribution and the directional spreading on the surface elevation of the focused wave, the maximum generated wave crest, and the shifting of the focusing point, are numerically investigated. Especially, the effects of the wave directionality on the focused wave properties are emphasized. The numerical results show that the shifting of the focusing point and the maximum crest of the wave group are dependent on the amplitude of the focused wave, the central frequency, and the wave amplitude distribution type. The wave directionality has a definite effect on multidirectional focused waves. Generally, it can even out the difference between the simulated wave amplitude and the amplitude expected from theory and reduce the shifting of the focusing points, implying that the higher order interaction has an influence on wave focusing, especially for 2D wave. In 3D wave groups, a broader directional spreading weakens the higher nonlinear interactions.

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.

GENERALIZED FINITE SPECTRAL METHOD FOR 1D BURGERS AND KDV EQUATIONS

A generalized finite spectral method is proposed. The method is of highorder accuracy. To attain high accuracy in time discretization, the fourth-order AdamsBashforth-Moulton predictor and corrector scheme was used. To avoid numerical oscillations caused by the dispersion term in the KdV equation, two numerical techniques were introduced to improve the numerical stability. The Legendre, Chebyshev and Hermite polynomials were used as the basis functions. The proposed numerical scheme is validated by applications to the Burgers equation （nonlinear convection- diffusion problem） and KdV equation（single solitary and 2-solitary wave problems）, where analytical solutions are available for comparison. Numerical results agree very well with the corresponding analytical solutions in all cases.

SPECTRAL METHOD IN TIME FOR KdV EQUATIONS

This paper presents a fully spectral discretization method for solving KdV equations with periodic boundary conditions.Chebyshev pseudospectral approximation in the time direction and Fourier Galerkin approximation in the spatial direction.The expansion coefficients are determined by minimizing an object funictional.Rapid convergence of the method is proved.

Spectral Method for a Class of Cahn-Hilliard Equation with Nonconstant Mobility

In this paper, we propose and analyze a full-discretization spectral approximation for a class of Cahn-Hilliard equation with nonconstant mobility. Convergenee analysis and error estimates are presented and numerical experiments are carried out.

Spectral Method for Solving Time Dependent Flow of Upper-Convected Maxwell Fluid in Tube

The ti me dependent flow of upper-convected Maxwell fluid in a horizontal circular pip e is studied by spectral method. The time dependent problem is mathematically re duced to a partial differential equation of second order. By using spectral meth od the partial differential equation can be reduced to a system of ordinary diff erential equations for different terms of Chebyshev polynomials approximations. The ordinary differential equations are solved by Laplace transform and the eige nvalue method that leads to an analytical form of the solutions.

Spectral Method for Three-Dimensional Nonlinear Klein-Gordon Equation by Using Generalized Laguerre and Spherical Harmonic Functions

In this paper,a generalized Laguerre-spherical harmonic spectral method is proposed for the Cauchy problem of three-dimensional nonlinear Klein-Gordon equation. The goal is to make the numerical solutions to preserve the same conservation as that for the exact solution.The stability and convergence of the proposed scheme are proved.Numerical results demonstrate the efficiency of this approach.We also establish some basic results on the generalized Laguerre-spherical harmonic orthogonal approximation,which play an important role in spectral methods for various problems defined on the whole space and unbounded domains with spherical geometry.

Chebyshev finite spectral method with extended moving grids

A Chebyshev finite spectral method on non-uniform meshes is proposed. An equidistribution scheme for two types of extended moving grids is used to generate grids. One type is designed to provide better resolution for the wave surface, and the other type is for highly variable gradients. The method has high-order accuracy because of the use of the Chebyshev polynomial as the basis function. The polynomial is used to interpolate the values between the two non-uniform meshes from a previous time step to the current time step. To attain high accuracy in the time discretization, the fourth-order Adams-Bashforth-Moulton predictor and corrector scheme is used. To avoid numerical oscillations caused by the dispersion term in the Korteweg-de Vries （KdV） equation, a numerical technique on non-uniform meshes is introduced. The proposed numerical scheme is validated by the applications to the Burgers equation （nonlinear convectiondiffusion problems） and the KdV equation （single solitary and 2-solitary wave problems）, where analytical solutions are available for comparisons. Numerical results agree very well with the corresponding analytical solutions in all cases.

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.

Mixed spectral method for exterior problems of Navier-Stokes equations by using generalized Laguerre functions

In this paper, we investigate the mixed spectral method using generalized Laguerre functions for exterior problems of fourth order partial differential equations. A mixed spectral scheme is provided for the stream function form of the Navier-Stokes equations outside a disc. Numerical results demonstrate the spectral accuracy in space.

3-D simulations of freak waves based on high-order spectral method

Three-dimensional ( 3-D) directional wave focusing is one of the mechanisms that contribute to the generation of freak waves. To simulate and analyze this phenomenon,a 3-D wave focusing model is proposed based on the enhanced high-order spectral method,which solves the fully nonlinear potential flow equations with a free surface within periodic unbounded 3-D domains. The numerical model is validated against a fifth-order Stokes solution for regular waves. Laboratory-scale freak waves are observed with wave components having equal amplitudes. Investigations of the appearance and propagation of freak-wave events in a 3-D open wavefield defined by a directional wave spectrum are then realized.

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.

Fully discrete Jacobi-spherical harmonic spectral method for Navier-Stokes equations

A fully discrete Jacobi-spherical harmonic spectral method is provided for the Navier-Stokes equations in a ball. Its stability and convergence are proved. Numerical results show efficiency of this approach. The proposed method is also applicable to other problems in spherical geometry.

Evaluation of Double Average Asian Options by the Legendre Spectral Method

In this paper, the evaluation of discretely sampled Asian options was considered by numerically solving the associated partial differential equations with the Legendre spectral method. Double average options were discussed as examples. The problem is a parabolic one on a finite domain whose equation degenerates into ordinary differential equations on the boundaries. A fully discrete scheme was established by using the Legendre spectral method in space and the Crank-Nicolson finite difference scheme in time. The stability and convergence of the scheme were analyzed. Numerical results show that the method can keep the spectral accuracy in space for such degenerate problems.

Spectral Method for Semilinear Parabolic Integrodifferential Equations

Based on the discussion of the semidiscretization of a parabolic equation with asemilinear memory term,an error estimate is derived for the fully discrete scheme with spectral method in space and the backward Euler method in time The trapezoidal rule is adopted.for the quadrature of the memory term and the quadrature error isestimated.

Application of Galerkin spectral method for tearing mode instability

Magnetic reconnection and tearing mode instability play a critical role in many physical processes.The application of Galerkin spectral method for tearing mode instability in two-dimensional geometry is investigated in this paper.A resistive magnetohydrodynamic code is developed,by the Galerkin spectral method both in the periodic and aperiodic directions.Spectral schemes are provided for global modes and local modes.Mode structures,resistivity scaling,convergence and stability of tearing modes are discussed.The effectiveness of the code is demonstrated,and the computational results are compared with the results using Galerkin spectral method only in the periodic direction.The numerical results show that the code using Galerkin spectral method individually allows larger time step in global and local modes simulations,and has better convergence in global modes simulations.

A Spectral Method for Convection-Diffusion Equations

In the practical problems such as nuclear waste pollution and seawater intrusion etc., many problems are reduced to solving the convection-diffusion equation, so the research of convection-diffusion equation is of great value. In this work, a spectral method is presented for solving one and two dimensional convection-diffusion equation with source term. The finite difference method is also used to solve the convection diffusion equation. The numerical experiments show that the spectral method is more efficient than other methods for solving the convection-diffusion equation.

Efficient Finite Difference/Spectral Method for the Time Fractional Ito Equation Using Fast Fourier Transform Technic

A finite difference/spectral scheme is proposed for the time fractional Ito equation.The mass conservation and stability of the numerical solution are deduced by the energy method in the L^(2)norm form.To reduce the computation costs,the fast Fourier transform technic is applied to a pair of equivalent coupled differential equations.The effectiveness of the proposed algorithm is verified by the first numerical example.The mass conservation property and stability statement are confirmed by two other numerical examples.

Conservative Discontinuous Galerkin/Hermite Spectral Method for the Vlasov-Poisson System

We propose a class of conservative discontinuous Galerkin methods for the Vlasov-Pois-son system written as a hyperbolic system using Hermite polynomials in the velocity vari-able.These schemes are designed to be systematically as accurate as one wants with prov-able conservation of mass and possibly total energy.Such properties in general are hard to achieve within other numerical method frameworks for simulating the Vlasov-Poisson system.The proposed scheme employs the discontinuous Galerkin discretization for both the Vlasov and the Poisson equations,resulting in a consistent description of the distribu-tion function and the electric field.Numerical simulations are performed to verify the order of the accuracy and conservation properties.

A SPECTRAL METHOD FOR A WEAKLY SINGULAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATION WITH PANTOGRAPH DELAY

In this paper,a Jacobi-collocation spectral method is developed for a Volterraintegro-differential equation with delay,which contains a weakly singular kernel.We use a function transformation and a variable transformation to change the equation into a new Volterra integral equation defined on the standard interval[-1,1],so that the Jacobi orthogonal polynomial theory can be applied conveniently.In order to obtain high order accuracy for the approximation,the integral term in the resulting equation is approximated by Jacobi spectral quadrature rules.In the end,we provide a rigorous error analysis for the proposed method.The spectral rate of convergence for the proposed method is established in both the L^(∞)-norm and the weighted L^(2)-norm.