On stability of numerical schemes via frozen coefficients and the magnetic induction equations

We study finite difference discretizations of initial boundary value problems for linear symmetric hyperbolic systems of equations in multiple space dimensions. The goal is to prove stability for SBP-SAT (Summation by Parts—Simultaneous Approximation Term) finite difference schemes for equations with variable coefficients. We show stability by providing a proof for the principle of frozen coefficients, i.e., showing that variable coefficient discretization is stable provided that all corresponding constant coefficient discretizations are stable.     

Symmetric hyperbolic difference schemes and matrix problems

Sufficient conditions for the stability of multidimensional finite difference schemes are difficult to obtain. It is shown that for special families of amplification matrices G (A, B) a sufficient condition for power boundedness can be obtained by replacing the matrices by appropriate scalars, and so the problem is reduced to a scalar one. As one application it is shown that the Lax-Wendroff scheme in two dimensions is stable if |Au|^{$\text{2}\text{3}$} + |Bu|^{$\text{2}\text{3}$} ? 1 for all real unit vectors u. The Lax- Wendroff scheme with stabilizer does not always permit such large time steps. It is conjectured that the analysis for all symmetric hyperbolic schemes can be reduced to the scalar case.     

On the stability of a class of noncooperative games

The Nash equilibrium of a class of games generated from a market is examined. Demands are assumed linear, and production constraints are imposed. The equilibrium is shown to be solvable as a complementarity problem. If the demand matrix is a positive definite symmetric z-matrix, then the Nash equilibrium is stable. If the demand matrix is not symmetric, an additional condition yielding stability is developed.     

Generating Polynomials and Symmetric Tensor Decompositions

This paper studies symmetric tensor decompositions. For symmetric tensors, there exist linear relations of recursive patterns among their entries. Such a relation can be represented by a polynomial, which is called a generating polynomial. The homogenization of a generating polynomial belongs to the apolar ideal of the tensor. A symmetric tensor decomposition can be determined by a set of generating polynomials, which can be represented by a matrix. We call it a generating matrix. Generally, a symmetric tensor decomposition can be determined by a generating matrix satisfying certain conditions. We characterize the sets of such generating matrices and investigate their properties (e.g., the existence, dimensions, nondefectiveness). Using these properties, we propose methods for computing symmetric tensor decompositions. Extensive examples are shown to demonstrate the efficiency of proposed methods.     

Block matrices and symmetric perturbations

We prove that if A=[A_ij]∈R^N,N is a block symmetric matrix and y is a solution of a nearby linear system (A+E)y=b, then there exists F=F^T such that y solves a nearby symmetric system (A+F)y=b, if A is symmetric positive definite or the matricial norm μ(A)=(‖A_ij‖₂) is diagonally dominant. Our blockwise analysis extends existing normwise and componentwise results on preserving symmetric perturbations (cf. [J.R. Bunch, J.W. Demmel, Ch. F. Van Loan, The strong stability of algorithms for solving symmetric linear systems, SIAM J.Matrix Anal. Appl. 10 (4) (1989) 494-499; D. Herceg, N. Kreji?, On the strong componentwise stability and H-matrices, Demonstratio Mathematica 30 (2) (1997) 373-378; A. Smoktunowicz, A note on the strong componentwise stability of algorithms for solving symmetric linear systems, Demonstratio Mathematica 28 (2) (1995) 443-448]).     

Optimizing Lyapunov functions for systems with structured uncertainties

The robust stability problem of a nominally linear system with nonlinear, time-varying structured perturbationsp _j ,j=1,...,q, is considered. The system is of the form $$\dot x = A_N x + \sum\limits_{j = 1}^q {p_j A_j x} .$$ When the Lyapunov direct method is utilized to solve the problem, the most frequently chosen Lyapunov function is some quadratic form. The paper presents a procedure of optimization of Lyapunov functions. Under some simple conditions, the weak convergence of the procedure is ensured, making the procedure effective in solving the robust stability problem. The procedure is simple, requiring only numerical routines such as inverting positive-definite symmetric matrices and determining the eigenvalues and eigenvectors of symmetric matrices. It is expected that the optimal Lyapunov function may be used in a robust linear feedback controller design. The examples demonstrate the effectiveness of the method. As shown when considering a system of dimension 24, the method is effective for large-scale systems.     

Linear instability of relative equilibria for n-body problems in the plane

Following Smale, we study simple symmetric mechanical systems of n point particles in the plane. In particular, we address the question of the linear and spectral stability properties of relative equilibria, which are special solutions of the equations of motion.     

Eigenvalues and eigenvectors of symmetric centrosymmetric matrices

It is proved that the eigenvectors of a symmetric centrosymmetric matrix of order N are either symmetric or skew symmetric, and that there are ?N/2? symmetric and ?N/2? skew symmetric eigenvectors. Some previously known but widely scattered facts about symmetric centrosymmetric matrices are presented for completeness. Special cases are considered, in particular tridiagonal matrices of both odd and even order, for which it is shown that the eigenvectors corresponding to the eigenvalues arranged in descending order are alternately symmetric and skew symmetric provided the eigenvalues are distinct.     

The transonic shock in a nozzle, 2-D and 3-D complete Euler systems

In this paper, we study a transonic shock problem for the Euler flows through a class of 2-D or 3-D nozzles. The nozzle is assumed to be symmetric in the diverging (or converging) part. If the supersonic incoming flow is symmetric near the divergent (or convergent) part of the nozzle, then, as indicated in Section 147 of [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publ., New York, 1948], there exist two constant pressures P₁ and P₂ with P₁<P₂ such that for given constant exit pressure Pe∈(P₁,P₂), a symmetric transonic shock exists uniquely in the nozzle, and the position and the strength of the shock are completely determined by Pe. Moreover, it is shown in this paper that such a transonic shock solution is unique under the restriction that the shock goes through the fixed point at the wall in the multidimensional setting. Furthermore, we establish the global existence, stability and the long time asymptotic behavior of an unsteady symmetric transonic shock under the exit pressure Pe when the initial unsteady shock lies in the symmetric diverging part of the 2-D or 3-D nozzle. On the other hand, it is shown that an unsteady symmetric transonic shock is structurally unstable in a global-in-time sense if it lies in the symmetric converging part of the nozzle.     

Exponential stability criteria of uncertain systems with multiple time delays

The exponential stability (with convergence rate α) of uncertain linear systems with multiple time delays is studied in this paper. Using the characteristic function of linear time-delay system, stability criteria are derived to guarantee α-stability. Sufficient conditions are also obtained for exponential stability of uncertain parametric systems with multiple time delays. For two-dimensional time-invariant system with multiple time delays, the proposed stability criteria are shown to be less conservative than those in the literature. Numerical examples are given to illustrate the validity of our new stability criteria.     

Long-time behavior of the difference solutions to the 2D GKS equation

We study the long-time behavior of the finite difference solution to the generalized Kuramoto-Sivashinsky equation in two space dimensions with periodic boundary conditions. The unique solvability of numerical solution is shown. It is proved that there exists a global attractor of the discrete dynamical system and the upper semicontinuity d(A_h,τ,A)→0. Finally, we obtain the long-time stability and convergence of the difference scheme. Our results show that the difference scheme can effectively simulate the infinite dimensional dynamical systems.     

Entropic structure of multicomponent reactive flows with partial equilibrium reduced chemistry

In this paper, we investigate the system of equations modelling multicomponent reactive flows with detailed transport and complex chemistry in the limit of partial equilibrium. The reduced system is obtained using a projection step compatible with the chemical entropy production. The reduced multicomponent transport and convection fluxes are shown to be compatible with the mathematical entropy thus providing a symmetric form as well as normal forms for the reduced system. This yields global existence and asymptotic stability around constant equilibrium states for the Cauchy problem on the partial equilibrium manifold in all space dimensions. Copyright © 2004 John Wiley & Sons, Ltd.     

Canonical decompositions of symmetric submodular systems

Let E be a finite set, R be the set of real numbers and f: 2^E→R be a symmetric submodular function. The pair (E,f) is called a symmetric submodular system. We examine the structures of symmetric submodular systems and provide a decomposition theory of symmetric submodular systems. The theory is a generalization of the decomposition theory of 2-connected graphs developed by Tutte and can be applied to any (symmetric) submodular systems.     

Linear maps preserving commutativity

Commutativity-preserving maps on the real space of all real symmetric or complex self-adjoint matrices are characterized. Related results are given for adjoint-preserving maps defined on all n × n matrices. These results are extended to infinite dimensions in the case of invertible maps.     

Majorants of the Dirichlet kernels and the Dini pointwise tests for generalized Haar systems

In this paper, we obtain five tests (three of which are symmetric) of pointwise convergence of Fourier series with respect to generalized Haar systems; the tests are similar to the Dini convergence tests. It is shown that the Dini convergence tests for Price systems are also valid for generalized Haar systems. It is also shown that the classicalDini convergence test does not apply, in general, even to generalized Haar systems, although the classical symmetric Dini test for generalized Haar systems is valid. Also upper bounds for the Dirichlet kernels for generalized Haar systems are obtained.     

Solution of symmetric linear complementarity problems by iterative methods

A unified treatment is given for iterative algorithms for the solution of the symmetric linear complementarity problem: $$Mx + q \geqslant 0, x \geqslant 0, x^T (Mx + q) = 0$$ , whereM is a givenn×n symmetric real matrix andq is a givenn×1 vector. A general algorithm is proposed in which relaxation may be performed both before and after projection on the nonnegative orthant. The algorithm includes, as special cases, extensions of the Jacobi, Gauss-Seidel, and nonsymmetric and symmetric successive over-relaxation methods for solving the symmetric linear complementarity problem. It is shown first that any accumulation point of the iterates generated by the general algorithm solves the linear complementarity problem. It is then shown that a class of matrices, for which the existence of an accumulation point that solves the linear complementarity problem is guaranteed, includes symmetric copositive plus matrices which satisfy a qualification of the type: $$Mx + q > 0 for some x in R^n $$ . Also included are symmetric positive-semidefinite matrices satisfying this qualification, symmetric, strictly copositive matrices, and symmetric positive matrices. Furthermore, whenM is symmetric, copositive plus, and has nonzero principal subdeterminants, it is shown that the entire sequence of iterates converges to a solution of the linear complementarity problem.     

Gauss-Runge-Kutta-Nyström methods

Symplectic Runge-Kutta-Nyström methods are frequently used to integrate secondorder systems of the special formÿ=f(y), where the functionf is the gradient of a scalar field multiplied by a regular matrix. In this paper Gauss-Runge-Kutta-Nyström methods, i.e., methods of the highest order, are discussed. It is proved that these methods are always symmetric and that symmetry is equivalent to symplecticness. Furthermore, it is shown that for each stage number the symplectic Gauss-Runge-Kutta-Nyström methods are given by a family of methods with one free parameter.     

Quaternion solution for the rock’n’roller: Box orbits, loop orbits and recession

We consider two types of trajectories found in a wide range of mechanical systems, viz. box orbits and loop orbits. We elucidate the dynamics of these orbits in the simple context of a perturbed harmonic oscillator in two dimensions. We then examine the small-amplitude motion of a rigid body, the rock’n’roller, a sphere with eccentric distribution of mass. The equations of motion are expressed in quaternionic form and a complete analytical solution is obtained. Both types of orbit, boxes and loops, are found, the particular form depending on the initial conditions. We interpret the motion in terms of epi-elliptic orbits. The phenomenon of recession, or reversal of precession, is associated with box orbits. The small-amplitude solutions for the symmetric case, or Routh sphere, are expressed explicitly in terms of epicycles; there is no recession in this case.     

Global existence for three‐dimensional incompressible isotropic elastodynamics

The existence of global‐in‐time classical solutions to the Cauchy problem for incompressible, nonlinear, isotropic elastodynamics for small initial displacements is proved. The generalized energy method is used to obtain strong dispersive estimates that are needed for long‐time stability. This requires the use of weighted local decay estimates for the linearized equations, which are obtained as a special case of a new general result for certain isotropic symmetric hyperbolic systems. In addition, the pressure that arises as a Lagrange multiplier to enforce the incompressibility constraint is estimated as a nonlinear term. The incompressible elasticity equations are inherently linearly degenerate in the isotropic case; i.e., the equations satisfy a null condition necessary for global existence in three dimensions. © 2007 Wiley Periodicals, Inc.     

Quasi-residual quasi-symmetric designs

It is shown that for each λ ? 3, there are only finitely many quasi-residual quasi-symmetric (QRQS) designs and that for each pair of intersection numbers (x, y) not equal to (0, 1) or (1, 2), there are only finitely many QRQS designs.A design is shown to be affine if and only if it is QRQS with x = 0. A projective design is defined as a symmetric design which has an affine residual. For a projective design, the block-derived design and the dual of the point-derivate of the residual are multiples of symmetric designs.