A powerful method for constructing difference families and optimal optical orthogonal codes

In this paper we proceed in the way indicated by R. M. Wilson for obtaining simple difference families from finite fields [28]. We present a theorem which includes as corollaries all the known direct techniques based on Galois fields, and provides a very effective method for constructing a lot of new difference families and also new optimal optical orthogonal codes.By means of our construction—just to give an idea of its power—it has been established that the only primesp<10⁵ for which the existence of a cyclicS(2, 9,p) design is undecided are 433 and 1009. Moreover we have considerably improved the lower bound on the minimumv for which anS(2, 15,v) design exists.     

An alternating direction implicit method for orthogonal spline collocation linear systems

Summary An Alternating Direction Implicit method is analyzed for the solution of linear systems arising in high-order, tensor-product orthogonal spline collocation applied to some separable, second order, linear, elliptic partial differential equations in rectangles. On anNxN partition, with Jordan's selection of the acceleration parameters, the method requiresO(N ² ln ² N) arithmetic operations to produce an approximation whose accuracy, in theH ¹-norm, is that of the collocation solution.     

An efficient method for solving difference systems for elliptic differential equations

A method for solving systems of linear algebraic equations arising in connection with the approximation of boundary value problems for elliptic partial differential equations is proposed. This method belongs to the class of conjugate directions method applied to a preliminary transformed system of equations. A model example is used to explain the idea underlying this method and to investigate it. Results of numerical experiments that confirm the method’s efficiency are discussed.     

Some reduced finite difference schemes based on a proper orthogonal decomposition technique for parabolic equations

In this paper, some reduced finite difference schemes based on a proper orthogonal decomposition (POD) technique for parabolic equations are derived. Also the error estimates between the POD approximate solutions of the reduced finite difference schemes and the exact solutions for parabolic equations are established. It is shown by considering the results of two numerical examples that the numerical results are consistent with theoretical conclusions. Moreover, it is also shown that the POD reduced finite difference schemes are feasible and efficient.     

The upwind finite difference fractional steps method for nonlinear coupled systems

For nonlinear coupled system of multilayer dynamics of fluids in porous media, a second‐order upwind finite‐difference fractional‐steps scheme applicable to parallel arithmetic are put forward, and two‐ and three‐dimensional schemes are used to form a complete set. Some techniques, such as calculus of variations, multiplicative commutation rule of difference operators, decomposition of high‐order difference operators, and prior estimates are adopted. Optimal order estimates in l² norm are derived to determine the error in the approximate solution. This method has already been applied to the numerical simulation of migration‐accumulation of oil resources. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Finite difference method for a combustion model

We study a projection and upwind finite difference scheme for a combustion model problem. Convergence to weak solutions is proved under the Courant-Friedrichs-Lewy condition. More assumptions are given on the ignition temperature; then convergence to strong detonation wave solutions or to weak detonation wave solutions is proved.

     

General method for producing a boundary-fitted orthogonal curvilinear mesh

A general method is described for computing an orthogonal mesh fitted to a two-dimensional physical domain with arbitrary closed boundary. The method allows optimum control of mesh spacing through the introduction of arbitrary (with weak constraints) ‘packing’ functions into the elliptic governing equations. Two particular aspects are addressed: first, the condition on a scaling factor which normalizes the mesh aspect ratio; second, the condition for avoiding run-out of the mesh beyond the boundaries of the physical domain.Conversion of the equations to finite difference form and appropriate iterative techniques are discussed. Finally applications of the method in the context of flow across a bundle of rods are presented.     

Model recovery for Hammerstein systems using the auxiliary model based orthogonal matching pursuit method

This article investigates parameter and order identification of a block-oriented Hammerstein system by using the orthogonal matching pursuit method in the compressive sensing theory which deals with how to recover a sparse signal in a known basis with a linear measurement model and a small set of linear measurements. The idea is to parameterize the Hammerstein system into the linear measurement model containing a measurement matrix with some unknown variables and a sparse parameter vector by using the key variable separation principle, then an auxiliary model based orthogonal matching pursuit algorithm is presented to recover the sparse vector.The standard orthogonal matching pursuit algorithm with a known measurement matrix is a popular recovery strategy by picking the supporting basis and the corresponding non-zero element of a sparse signal in a greedy fashion. In contrast to this, the auxiliary model based orthogonal matching pursuit algorithm has unknown variables in the measurement matrix. For a K-sparse signal, the standard orthogonal matching pursuit algorithm takes a fixed number of K stages to pick K columns (atoms) in the measurement matrix, while the auxiliary model based orthogonal matching pursuit algorithm takes steps larger than K to pick K atoms in the measurement matrix with the process of picking and deleting atoms, due to the gradually accurate estimates of the unknown variables step by step.The auxiliary model based orthogonal matching pursuit algorithm can simultaneously identify parameters and orders of the Hammerstein system, and has a high efficient identification performance.     

BEHAVIOR OF SOLUTIONS FOR A CLASS OF DIFFERENCE SYSTEMS

In this paper, the discrete-time neural network model of two neurons with piecewise constant argument is considered. Some sufficient conditions under which every solution is either periodic or convergent are obtained.     

On a Numerical-Analytic method for second-order difference equations

For second-order difference equations, we justify the scheme of the Samoilenko numerical-analytic method for finding periodic solutions.     

Finite difference scheme based on proper orthogonal decomposition for the nonstationary Navier-Stokes equations

The proper orthogonal decomposition(POD)and the singular value decomposition(SVD) are used to study the finite difference scheme(FDS)for the nonstationary Navier-Stokes equations. Ensembles of data are compiled from the transient solutions computed from the discrete equation system derived from the FDS for the nonstationary Navier-Stokes equations.The optimal orthogonal bases are reconstructed by the elements of the ensemble with POD and SVD.Combining the above procedures with a Galerkin projection approach yields a new optimizing FDS model with lower dimensions and a high accuracy for the nonstationary Navier-Stokes equations.The errors between POD approximate solutions and FDS solutions are analyzed.It is shown by considering the results obtained for numerical simulations of cavity flows that the error between POD approximate solution and FDS solution is consistent with theoretical results.Moreover,it is also shown that this validates the feasibility and efficiency of POD method.     

Higher-order compact finite difference method for systems of reaction–diffusion equations

This paper is concerned with a compact finite difference method for solving systems of two-dimensional reaction–diffusion equations. This method has the accuracy of fourth-order in both space and time. The existence and uniqueness of the finite difference solution are investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear term. Three monotone iterative algorithms are provided for solving the resulting discrete system efficiently, and the sequences of iterations converge monotonically to a unique solution of the system. A theoretical comparison result for the various monotone sequences is given. The convergence of the finite difference solution to the continuous solution is proved, and Richardson extrapolation is used to achieve fourth-order accuracy in time. An application is given to an enzyme–substrate reaction–diffusion problem, and some numerical results are presented to demonstrate the high efficiency and advantages of this new approach.     

On the falsity of a conjecture on orthogonal steiner triple systems

A pair of orthogonal Steiner triple systems of order ν = 27 is constructed, thus showing the conjecture about the non-existence of a pair of orthogonal Steiner triple systems of orders ν ≡ 3 (mod 6) to be false.     

Weyl-Titchmarsh theory for symplectic difference systems

In this work, we establish Weyl-Titchmarsh theory for symplectic difference systems. This paper extends classical Weyl-Titchmarsh theory and provides a foundation for studying spectral theory of symplectic difference systems.