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.

A Regional Spectral Nested Multilevel Primitive Equation Model

By means of vertical normal modes a regional nested multilevel primitive equation model can be reduced to several sets of shallow water equations characterized by various equivalent depths. Therefore, time integration of the model in spectral form can be performed in the manner similar to those used in the spectral nested shallow water equation model case.

A Regional Spectral Nested Shallow Water Equation Model

A method to expand meteorological elements in terms of finite double Fourier series in a limited-region and a spectral nested shallow water equation model based upon the method with conformal map projection in rectangular coordinates, have been proposed, and computational stability and efficiency of time integration have been discussed.

THE FORMULATION OF FIDELITY SCHEMES OF PHYSICAL CONSERVATION LAWS AND IMPROVEMENTS ON A TRADITIONAL SPECTRAL MODEL OF BAROCLINIC PRIMITIVE EQUATIONS FOR NUMERICAL PREDICTION

In this paper,two formulation theorems of time-difference fidelity schemes for general quadratic and cubic physical conservation laws are respectively constructed and proved,with earlier major conserving time-discretized schemes given as special cases.These two theorems can provide new mathematical basis for solving basic formulation problems of more types of conservative time- discrete fidelity schemes,and even for formulating conservative temporal-spatial discrete fidelity schemes by combining existing instantly conserving space-discretized schemes.Besides.the two theorems can also solve two large categories of problems about linear and nonlinear computational instability. The traditional global spectral-vertical finite-difference semi-implicit model for baroclinic primitive equations is currently used in many countries in the world for operational weather forecast and numerical simulations of general circulation.The present work,however,based on Theorem 2 formulated in this paper,develops and realizes a high-order total energy conserving semi-implicit time-difference fidelity scheme for global spectral-vertical finite-difference model of baroclinic primitive equations.Prior to this,such a basic formulation problem remains unsolved for long,whether in terms of theory or practice.The total energy conserving semi-implicit scheme formulated here is applicable to real data long-term numerical integration. The experiment of thirteen FGGE data 30-day numerical integration indicates that the new type of total energy conserving semi-implicit fidelity scheme can surely modify the systematic deviation of energy and mass conserving of the traditional scheme.It should be particularly noted that,under the experiment conditions of the present work,the systematic errors induced by the violation of physical laws of conservation in the time-discretized process regarding the traditional scheme designs(called type Z errors for short)can contribute up to one-third of the total systematic root-mean-square(RMS)error at the end of second week of the integration and exceed one half of the total amount four weeks afterwards.In contrast,by realizing a total energy conserving semi-implicit fidelity scheme and thereby eliminating corresponding type Z errors, roughly an average of one-fourth of the RMS errors in the traditional forecast cases can be reduced at the end of second week of the integration,and averagely more than one-third reduced at integral time of four weeks afterwards.In addition,experiment results also reveal that,in a sense,the effects of type Z errors are no less great than that of the real topographic forcing of the model.The prospects of the new type of total energy conserving fidelity schemes are very encouraging.

THE LARGE TIME CONVERGENCE OF SPECTRAL METHOD FOR GENERALIZED KURAMOTO-SIVASHINSKY EQUATIONS

In this paper we use the spectral method to analyse the generalized Kuramoto-Sivashinsky equations. We prove the existence and uniqueness of global smooth solution of the equations. Finally, we obtain the error estimation between spectral approximate solution and exact solution on large time.

FOURIER-LEGENDRE SPECTRAL METHOD FOR THE UNSTEADY NAVIER-STOKES EQUATIONS

Fourier-Legendre spectral approximation for the unsteady Navier-Stokes equations is analyzed. The generalized stability and convergence are proved respectively.

A SPECTRAL METHOD FOR A CLASS OF NONLINEAR QUASI-PARABOLIC EQUATIONS

In this paper, we consider the numerical solution of quasi-parabolic equations of higher order by a spectral method, and propose a computational formula. We give an error estimate of approximate solutions, and prove the convergence of the approximate method and numerical stability on initial values. Under certain conditions, which are much weaker than the conditions in [6], we gain the same convergence rate as in [6].

SPECTRAL METHOD FOR ZAKHAROV EQUATION WITH PERIODIC BOUNDARY CONDITIONS

This paper describes the spectral method for numerically solving Zakharov equation with periodicboundary conditions. This method is spectral method for spatial variable and difference method fortime variable. We make error estimation of approximate solution and prove the convergence of spectralmethod. We had given the convergence rate. Also, we prove the stability of approximate method forinitial values.

A Simplified Full-Wave Analysis for Microstrip Patch Antennas

The spectral domain integral equation(SDIE) provides an accurate and efficient method for computing the resonant frequency, radiation patterns, etc . Using continuous Fourier transform, the formulation utilizes the singular integral equations via the Glerkin's method to derive the deterministic equation with fewer mathematical manipulations. In contrast, discrete Fourier transform(DFT) requires intricate mathematical labor. The present scheme requires a small size, i.e ., (2×2) matrix, and it is possible to extract higher order modal solutions conveniently. Moreover, computation is reduced with the same convergence properties. Based on the present scheme, some results for resonant frequency and radiation patterns compared with available data and computed current distribution on the patch are presented.

An Efficient Spectral Method for Acoustic Scattering from Rough Surfaces

An efficient and accurate spectral method is presented for scattering problems with rough surfaces.A probabilistic framework is adopted by modeling the surface roughness as random process.An improved boundary perturbation technique is employed to transform the original Helmholtz equation in a random domain into a stochastic Helmholtz equation in a fixed domain.The generalized polynomial chaos(gPC)is then used to discretize the random space;and a Fourier-Legendre method to discretize the physical space.These result in a highly efficient and accurate spectral algorithm for acoustic scattering from rough surfaces.Numerical examples are presented to illustrate the accuracy and efficiency of the present algorithm.

Tightly coupled INS/CNS/spectral redshift integrated navigation system with the aid of redshift error measurement

The integration of an inertial navigation system(INS) and a celestial navigation system(CNS) has the superiority of high autonomy. However, its reliability and accuracy are permanently impaired under poor observation conditions. To address this issue, the present paper proposes a tightly coupled INS/CNS/spectral redshift(SRS) integration framework based on the spectral redshift error measurement. In the proposed method, a spectral redshift error measurement equation is investigated and embedded in the traditional tightly coupled INS/CNS integrated navigation system to achieve better anti-interference under complicated circumstances. Subsequently, the inaccurate redshift estimation from the low signal-to-noise ratio spectrum is considered in the integrated system, and an improved chi-square test-based covariance estimation method is incorporated in the federated Kalman filter, allowing to deal with measurement outliers caused by the inaccurate redshift estimation but not influencing the effect of other correct redshift measurements in suppressing the error of the navigation parameter on the filtering solution. Simulations and comprehensive analyses demonstrate that the proposed tightly coupled INS/CNS/SRS integrated navigation system can effectively handle outliers and outages under hostile observation conditions, resulting in improved performance.

Bernstein n-Widths for Classes of Convolution Functions with Kernels Satisfying Certain Oscillation Properties

In this paper, we consider some classes of 2π-periodic convolution functions Bp, and Kp with kernels having certain oscillation properties, which include the classical Sobolev class as special case. With the help of the spectral of nonlinear integral equations, we determine the exact values of Bernstein n-width of the classes Bp, Kp in the space Lp for 1 〈 p 〈 ∞.