Periodic solutions preserved by nonstandard finite-difference schemes for the Lotka–Volterra system: a different approach

The Lotka–Volterra predator–prey system x′ = x ? xy, y′ = ? y+xy is a good differential equation system for testing numerical methods. This model gives rise to mutually periodic solutions surrounding the positive fixed point (1,1), provided the initial conditions are positive. Standard finite-difference methods produce solutions that spiral into or out of the positive fixed point. Previously, the author [Roeger, J. Diff. Equ. Appl. 12(9) (2006), pp. 937–948], generalized three different classes of nonstandard finite-difference methods that when applied to the predator–prey system produced periodic solutions. These methods preserve weighted area; they are symplectic with respect to a noncanonical structure and have the property that the computed points do not spiral. In this paper, we use a different approach. We apply the Jacobian matrix procedure to find a fourth class of nonstandard finite-difference methods. The Jacobian matrix method gives more general nonstandard methods that also produce periodic solutions for the predator–prey model. These methods also preserve the positivity property of the solutions.     

On the solution of large-scale SDP problems by the modified barrier method using iterative solvers

The limiting factors of second-order methods for large-scale semidefinite optimization are the storage and factorization of the Newton matrix. For a particular algorithm based on the modified barrier method, we propose to use iterative solvers instead of the routinely used direct factorization techniques. The preconditioned conjugate gradient method proves to be a viable alternative for problems with a large number of variables and modest size of the constrained matrix. We further propose to avoid explicit calculation of the Newton matrix either by an implicit scheme in the matrix–vector product or using a finite-difference formula. This leads to huge savings in memory requirements and, for certain problems, to further speed-up of the algorithm. Dedicated to the memory of Jos Sturm.     

Ellipsoidal approximation of attainability sets of a linear system with indeterminate matrix

Linear dynamical systems described by finite-difference or differential equations are considered. It is assumed that the matrix of the system is either completely known or is subject to uncontrollable perturbations, so that each element is known only to within a certain possible interval. Outer approximations, by means of ellipsoids, are constructed for the attainability sets of such systems. The equations of evolution of the approximating ellipsoids are obtained. An example is presented.     

Robust stability and H∞ control for nonlinear discrete‐time switched systems with interval time‐varying delay

This paper considers the problems of the robust stability analysis and H_∞ controller synthesis for uncertain discrete‐time switched systems with interval time‐varying delay and nonlinear disturbances. Based on the system transformation and by introducing a switched Lyapunov‐Krasovskii functional, the novel sufficient conditions, which guarantee that the uncertain discrete‐time switched system is robust asymptotically stable are obtained in terms of linear matrix inequalities. Then, the robust H_∞ control synthesis via switched state feedback is studied for a class of discrete‐time switched systems with uncertainties and nonlinear disturbances. We designed a switched state feedback controller to stabilize asymptotically discrete‐time switched systems with interval time‐varying delay and H_∞ disturbance attenuation level based on matrix inequality conditions. Examples are provided to illustrate the advantage and effectiveness of the proposed method.     

Parallel Implementation of Stability Analysis of Difference Schemes with MATHEMATICA

We consider a parallel algorithm for investigating the stability of the schemes of the finite-difference and finite-volume methods that approximate the two-dimensional Euler equations of compressible fluid on a curvilinear grid. The algorithm is implemented with the aid of the computer algebra system Mathematica 3.0. We apply a two-level parallelization process. At the first level, the symbolic computation of the amplification matrix is parallelized by a parallel computation of the matrix rows on different processors. At the second level, the values of the coordinates of points of the stability-region boundary are computed numerically. For the communication between the workstations, we apply a special program, LaunchSlave, which uses the MathLink communication protocol. Examples of application of the proposed parallel symbolic/numerical algorithm are presented. Bibliography: 15 titles.     

Optimization of periodic systems

The problem of designing a regulator, optimal by a quadratic performance criterion, on an infinite time interval is examined for a linear periodic system. It is assumed that the control plant's motion is described by a system of linear periodic finite-difference equations. Controllable plants whose motion is described by differential and by finite-difference equations on different parts of the period are analyzed as well. The optimal regulator design problem is reduced to the determination of a periodic solution of an appropriate Riccati equation. An algorithm for constructing such a solution is derived. It is noted that this result can be used in periodic optimization problems /1/ and in the design of a stabilization system for a pacing apparatus.     

An exact global optimization method for deriving weights from pairwise comparison matrices

Some multiple-criteria decision making methods rank actions by associating weights to the different criteria or actions, which are pairwise compared via a positive reciprocal matrix A. There is a vast literature on proposals of different mathematical-programming methods to infer weights from such matrix A. However, it is seldom observed that such optimization problems may be multimodal, thus the standard local-search resolution techniques suggested may be trapped in local optima, yielding a wrong ranking of alternatives. In this note we show that standard tools of global optimization based on interval analysis, lead to globally optimal weights in reasonable time.     

Finite-difference method for the system of equations of tide dynamics

For the system of shallow water equations describing the tidal wave propagation, we construct a finite-difference scheme on an unstructured grid. We analyze the properties of the resulting system of equations; in particular, we show that, after the elimination of part of the variables, the matrix of the system becomes an M-matrix.     

Using Root Functions for Eigenvalue Problems of Ordinary Differential Operators

We study eigenvalue problems for an ordinary differential operator L acting on L ₂(?)-spaces (Problem 1) and on L ₂(J)-spaces (Problem 2). Here J is a bounded but large interval. Assuming that in Problem 1 the spectral parameter s lies in the set of normal points of L, we show that the structure of eigenspaces for both problems is similar to the structure of finite complex-valued matrices. In the case of a finite matrix, the geometry of eigenspaces is described by the Jordan form. In the case of ordinary differential operators, the corresponding geometry is described by a sequence of root functions. Therefore, the main tool of our studies is root functions for complex-valued analytical matrix functions.     

A family of methods for the wave equation in one- and two-space dimensions

A family of finite-difference methods is developed for the numerical solution of the simple wave equation. Local truncation errors are calculated for each member of the family and each is analyzed for stability. The concepts of A₀ stability and L₀ stability, well used in the literature on other types of partial differential equation, are discussed in relation to second-order hyperbolic equations. The numerical methods arc extended to cover two-dimensional wave equations and the methods developed in this article are tested on three problems from the literature.     

Compact high-order finite-element method for elliptic transport problems with variable coefficients

We present a new conforming bilinear Petrov-Galerkin finite-element scheme for elliptic transport problems with variable coefficients. This scheme combines a generalized test function and artificial diffusion to achieve O(h⁴) grid-point accuracy on uniform stencils of 3 × 3 in two dimensions without resorting to the extended stencils of high-order elements. The method is compared with upwind and high-order finite-difference schemes and the standard Galerkin finite-element method for representative test problems. © 1994 John Wiley & Sons, Inc.     

Application of the method of straight lines to determining the stressed-strained state and dynamic characteristics of a hollow noncircular cylinder

Using the method of straight lines, problems of elasticity theory for a noncircular hollow cylinder are investigated. By means of separation of variables along the generatrix and finite-difference approximation across the thickness, the input partial differential equations are reduced to a system of ordinary differential equations, which is solvable by the stable numerical method of discrete orthogonalization. Specific features of application of the method proposed to static and dynamic problems of a hollow noncircular cylinder are illustrated by way of examples. Bibliography: 5 titles. Translated from,Obchyslyuval’na ta Prykladna Matematyka, No. 76, 1992, pp. 89–87.     

High-order difference schemes for two-dimensional elliptic equations

A high-order finite-difference approximation is proposed for numerical solution of linear or quasilinear elliptic differential equation. The approximation is defined on a square mesh stencil using nine node points and has a truncation error of order h⁴. Several test problems, including one modeling convection-dominated flows, are solved using this and existing methods. The results clearly exhibit the superiority of the new approximation, in terms of both accuracy and computational efficiency.     

An analysis of alternating-direction methods for parabolic equations

Alternating-direction solution procedures for parabolic partial differential equations can be developed using finite-difference, finite-element, and collocation approximations in space. Each of these methods derives from a common alternating-direction formulation. Furthermore, each method leads to an O[(Δt)²] error which is in addition to the discretization error associated with standard multidimensional solutions. However, when dealing with equations having spatially varying coefficients, some alternating-direction formulations lead to yet other errors which are O(Δt). These latter errors, and thus the accuracy of the method, depend on the structure of the mass matrix associated with the approximating method.     

Fourth-order Finite-difference Methods for Two-point Boundary-value Problems

A number of new fourth-order accurate finite-difference methodsare developed for second-order ordinary differential equationsof the boundary-value type. Schemes are obtained for both linearand non-linear equations. In all cases, the solution of thedifference equations may be accomplished using a direct eliminationtechnique for linear tridiagonal matrix problems. The accuracyof the new methods is compared with existing finite-differencemethods on a theoretical basis as well as by considering a numberof example problems. It is concluded that the new methods offersignificant advantages for specific types of equations in termsof accuracy and/or computational efficiency.     

Delay-Dependent Robust Stabilization and <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Control for Nonlinear Stochastic Systems with Markovian Jump Parameters and Interval Time-Varying Delays

This paper considers the problems of delay-dependent robust stabilization and H _∞ control for nonlinear stochastic systems with Markovian jump parameters and interval time-varying delays. Based on the Lyapunov method and introducing some appropriate free-weighting matrices, sufficient conditions for the solvability of above problems have been investigated in terms of linear matrix inequalities (LMIs). Furthermore, the desired state feedback controller has also been designed by solving these LMIs. Finally, a numerical example is provided to demonstrate the potential of the proposed techniques.     

Reduced Vertex Set Result for Interval Semidefinite Optimization Problems

In this paper we propose a reduced vertex result for the robust solution of uncertain semidefinite optimization problems subject to interval uncertainty. If the number of decision variables is m and the size of the coefficient matrices in the linear matrix inequality constraints is n×n, a direct vertex approach would require satisfaction of 2 ^{n(m+1)(n+1)/2} vertex constraints: a huge number, even for small values of n and m. The conditions derived here are instead based on the introduction of m slack variables and a subset of vertex coefficient matrices of cardinality 2 ⁿ⁻¹, thus reducing the problem to a practically manageable size, at least for small n. A similar size reduction is also obtained for a class of problems with affinely dependent interval uncertainty. This work is supported by MIUR under the FIRB project “Learning, Randomization and Guaranteed Predictive Inference for Complex Uncertain Systems,” and by CNR RSTL funds.     

Numerical method for the KdV–Kawahara and Benney–Lin equations

We are concerned with the initial-boundary-value problem associated to the Korteweg – de Vries – Kawahara (KdVK) equation and Benney – Lin (BL) equation, which are transport equations perturbed by dispersive terms of 3rd and 5th order and a term of 4th order in the case of (BL) equation. These equations appear in several fluid dynamics problems. We obtain local smoothing effects that are uniform with respect to the size of the interval. We also propose a simple finite-difference scheme for the problem and prove its stability. Finally, we give some numerical examples. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The perturbed laplacian matrix of a graph

For a graph G, we define its perturbed Laplacian matrix as D?A(G) where A(G) is the adjacency matrix of G and D is an arbitrary diagonal matrix. Both the Laplacian matrix and the negative of the adjacency matrix are special instances of the perturbed Laplacian. Several well-known results, contained in the classical work of Fiedler and in more recent contributions of other authors are shown to be true, with suitable modifications, for the perturbed Laplacian. An appropriate generalization of the monotonicity property of a Fiedler vector for a tree is obtained. Some of the results are applied to interval graphs.     

Inverse problems for the matrix Sturm–Liouville equation with a Bessel-type singularity

The matrix Sturm–Liouville equation on a finite interval with a Bessel-type singularity in the end of the interval is studied. We consider inverse problems by the Weyl matrix and by the spectral data for this equation. Constructive solutions, based on the method of spectral mappings, are obtained for these inverse problems.