A One-step Method of Order 10 for y' = f(x, y)

In some situations, especially if one demands the solution ofthe differential equation with a great precision, it is preferableto use high-order methods. The methods considered here are similarto Runge—Kutta methods, but for the second-order equationy'= f(x, y). As for Runge—Kutta methods, the complexityof the order conditions grows rapidly with the order, so thatwe have to solve a non—linear system of 440 algebraicequations to obtain a tenth—order method. We demonstratehow this system can be solved. Finally we give the coefficients(20 decimals) of two methods with small local truncation errors.     

Blended extended linear multistep methods for the accurate numerical integration of stiff initial value problems

A class of blended extended linear multistep methods suitable for the approximate numerical integration of stiff systems of first order ordinary differential equations is described. These methods are formulated as a result of combining the second derivative extended backward differentiation formulae of Cash and the blended linear multistep methods of Skeel and Kong. The new methods combine a high order or accuracy with good stability properties and, as a direct consequence, they are often suitable for the numerical integration of stiff differential systems when high accuracy is requested. In the first part of the present paper we consider the derivation of these new blended methods and give the coefficients and stability regions for formulae of order up to and including 10. In the second half we consider their practical implementation. In particular we describe a variable order/variable step package based on these blended formulae and we evaluate the performance of this package on the well known DETEST test set. It is shown that the new code is reliable on this test set and is competitive with the well known second derivative method of Enright.     

Improving the Stability of Predictor-Corrector Methods by Residue Smoothing

Residue smoothing is usually applied in order to acceleratethe convergence of iteration processes. Here, we show that residuesmoothing can also be used in order to increase the stabilityregion of predictor-corrector methods. We shall concentrateon increasing the real stability boundary. The iteration parametersand the smoothing operators are chosen such that the stabilityboundary becomes as large as c(m, q)m²4^g where m is the numberof right-hand side evaluations per step, q the number of smoothingoperations applied to each right-hand side evaluation, and c(m,q) a slowly varying function of m and q, of magnitude 1.3 ina typical case. Numerical results show that, for a variety oflinear and nonlinear parabolic equations in one and two spatialdimensions, these smoothed predictor-corrector methods are atleast competitive with conventional implicit methods.     

Stability and Accuracy of Semi-discretized Finite Difference Methods

The relationship between the accuracy and stability of semi-discretizedfinite-difference schemes for the advection equation u_t = u_xis analysed. Given the scheme of accuracy p we establish that,subject to stability, p min {r+s+R+S, 2(r+R+1), 2(s+S)}. This is done by using the theory of order stars. Furthermore,we find stable methods which attain this bound for various choicesof r, s, R and S.     

High-accuracy Tridiagonal Finite Difference Approximations for Non-linear Two-point Boundary Value Problems

We discuss the construction of finite difference approximationsfor the non-linear two-point boundary value problem: y" = f(x,y), y(a)=A, y(b)=B. In the case of linear differential equations,the resulting finite difference schemes lead to tridiagonallinear systems. Approximations of orders higher than four involvederivatives of f. While several approximations of a particularorder are possible, we obtain the "simplest" of these approximationsleading to two high-accuracy methods of orders six and eight.These two methods are described and their convergence is established;numerical results are given to illustrate the order of accuracyachieved.     

An orthogonal spline collocation alternating direction implicit method for second-order hyperbolic problems

On a rectangular region, we consider a linear second-order hyperbolicinitial-boundary value problem involving a mixed derivativeterm, continuous variable coefficients and non-homogeneous Dirichletboundary conditions. In comparison to the alternating directionimplicit Laplace-modified method of Fernandes (1997), we formulateand analyse a new parameter-free alternating direction implicitscheme in which the standard central difference formula is usedfor the time approximation and orthogonal spline collocationis used for the spatial discretization. We establish unconditionalstability of the scheme, and its optimal order in the discretemaximum norm in time and the H¹ norm in space. Numerical experimentsindicate that the new scheme, which has the same order as themethod of Fernandes (1997, Numer. Math., 77, 223–241),is more accurate. We also show that the new scheme is easilygeneralized to the second-order hyperbolic problems on rectangularpolygons. Extensions of the scheme to problems with discontinuouscoefficients, nonlinear problems, and problems with other boundaryconditions are also discussed.     

Computing stability regions--Runge-Kutta methods for delay differential equations

We discuss the practical determination of stability regionswhen various fixed-stepsize Runge-Kutta (RK) methods, combinedwith continuous extensions, are applied to the linear delaydifferential equation (DDE) y'(t)= y(t)+µ(t–) (t) with fixed delay . It is significant that the delay is not limitedto an integer multiple of the stepsize, and that we considervarious continuous extensions. The stability loci obtained in practice indicate that the standardboundary-locus technique (BLT) can fail to map the RK DDE stabilityregion correctly. The aim of this paper is to present an alternativestability boundary algorithm that overcomes the difficultiesencountered using the BLT. This new algorithm can be used forboth explicit and implicit RK methods.     

The streamline-diffusion method for nonconforming Qrot1 elements on rectangular tensor-product meshes

When the streamline–diffusion finite element method isapplied to convection–diffusion problems using nonconformingtrial spaces, it has previously been observed that stabilityand convergence problems may occur. It has consequently beenproposed that certain jump terms should be added to the bilinearform to obtain the same stability and convergence behaviouras in the conforming case. The analysis in this paper showsthat for the Q^rot₁ 1 element on rectangular shape-regular tensor-productmeshes, no jump terms are needed to stabilize the method. Inthis case moreover, for smooth solutions we derive in the streamline–diffusionnorm convergence of order h^3/2 (uniformly in the diffusion coefficientof the problem), where h is the mesh diameter. (This estimateis already known for the conforming case.) Our analysis alsoshows that similar stability and convergence results fail tohold true for analogous piecewise linear nonconforming elements.     

Mean-square convergence of stochastic multi-step methods with variable step-size

We study mean-square consistency, stability in the mean-square sense and mean-square convergence of drift-implicit linear multi-step methods with variable step-size for the approximation of the solution of Itô stochastic differential equations. We obtain conditions that depend on the step-size ratios and that ensure mean-square convergence for the special case of adaptive two-step-Maruyama schemes. Further, in the case of small noise we develop a local error analysis with respect to the h$h$–ε$\epsilon$ approach and we construct some stochastic linear multi-step methods with variable step-size that have order 2 behaviour if the noise is small enough.     

A Hitchin-Kobayashi Correspondence for Coherent Systems on Riemann Surfaces

A coherent system (E, V) consists of a holomorphic bundle plusa linear subspace of its space of holomorphic sections. Motivatedby the usual notion in geometric invariant theory, a notionof slope stability can be defined for such objects. In the paperit is shown that stability in this sense is equivalent to theexistence of solutions to a certain set of gauge theoretic equations.One of the equations is essentially the vortex equation (thatis, the Hermitian–Einstein equation with an additionalzeroth order term), and the other is an orthonormality conditionon a frame for the subspace VH⁰(E).     

Conjugate Directions without Linear Searches

A modified form of the Quasi-Newton family of variable metricalgorithms used in function minimization is proposed that hasquadratic termination without requiring linear searches. Mostmembers of the Quasi-Newton family rely for quadratic terminationon the fact that with accurate linear searches the directionsgenerated, form a conjugate set when the function is quadratic.With some members of the family the convergence of the sequenceof approximate inverse Hessian matrices to the true inverseHessian is also stable. With the proposed modification the samesequence of matrices and the same set of conjugate directionsare generated without accurate linear searches. On a quadratic function the proposal is also related to a suggestionby Hestenes which generates the same set of conjugate directionswithout accurate linear searches. Both methods therefore findthe minimum of an n dimensional quadratic function in at mostn+2 function and gradient calls. On non-quadratic functions the proposal retains the main advantagesclaimed for both the stable Quasi-Newton and Hestenes approaches.It shows promise in that it is competitive with the most efficientunconstrained optimization algorithms currently available.     

On the Poles of Maximal Order of the Topological Zeta Function

The global and local topological zeta functions are singularityinvariants associated to a polynomial f and its germ at 0, respectively.By definition, these zeta functions are rational functions inone variable, and their poles are negative rational numbers.In this paper we study their poles of maximal possible order.When f is non-degenerate with respect to its Newton polyhedron,we prove that its local topological zeta function has at mostone such pole, in which case it is also the largest pole; wegive a similar result concerning the global zeta function. Moreover,for any f we show that poles of maximal possible order are alwaysof the form –1/N with N a positive integer. 1991 MathematicsSubject Classification 14B05, 14E15, 32S50.     

时间尺度上二阶线性动力学方程的一般解与Ulam稳定性

本文利用参数变易法研究了时间尺度上二阶变系数线性动力学方程的解与Ulam稳定性问题. 特别地,在不同的系数情形下建立了二阶常系数线性动力学方程的Ulam稳定性理论.     

Fixed and Minimal Order Compensators

Given a time-invariant linear plant, the authors consider aminimal (or fixed) order compensator such that the (unity feedback)closed loop is stable with respect to a given region in thecomplex plane. The so-called critical constraint is used tofind points in the (parameter space) stability region. Thisenables some practical constraints to be added in the compensatordesign.     

Convergence of a Finite-Difference Scheme for Second-Order Hyperbolic Equations with Variable Coefficients

We study the convergence of a finite-difference scheme for thefirst initial-boundary-value problem for linear second-orderhyperbolic equations with variable coefficients. Using the bilinearversion of the Bramble-Hilbert lemma we prove that the convergencerate in the discrete energy norm is of the order h^–2 ifthe exact solution belongs to the Sobolev space W₂(Q) with 2<<4.     

Multistep Discretization of Linear Hyperbolic Equations

Order and stability of multistep finite-difference discretizationsof the first-order linear hyperbolic equation u₁ = a(x)u_x areconsidered. We prove that if a stable method uses s upwind andrdownwind points and the coefficients depend only on the Courantnumber and on a(x) and its derivatives at the underlying gridpoint, then the order may not exceed r + s. This bound on orderis exactly half the bound of Strang & Iserles (1983) forconstant a. Furthermore, we prove that if r = s and a(x) isboth uniformly bounded and uniformly positive for x R thenthe new order barrier is attainable for every s 1.     

Asymptotic Mean-Square Stability of Two-Step Methods for Stochastic Ordinary Differential Equations

We deal with linear multi-step methods for SDEs and study when the numerical approximation shares asymptotic properties in the mean-square sense of the exact solution. As in deterministic numerical analysis we use a linear time-invariant test equation and perform a linear stability analysis. Standard approaches used either to analyse deterministic multi-step methods or stochastic one-step methods do not carry over to stochastic multi-step schemes. In order to obtain sufficient conditions for asymptotic mean-square stability of stochastic linear two-step-Maruyama methods we construct and apply Lyapunov-type functionals. In particular we study the asymptotic mean-square stability of stochastic counterparts of two-step Adams–Bashforth- and Adams–Moulton-methods, the Milne–Simpson method and the BDF method. AMS subject classification (2000) 60H35, 65C30, 65L06, 65L20     

Adaptive Lagrange-Galerkin methods for unsteady convection-diffusion problems

In this paper we derive an a posteriori error bound for the Lagrange-Galerkin discretisation of an unsteady (linear) convection-diffusion problem, assuming only that the underlying space-time mesh is nondegenerate. The proof of this error bound is based on strong stability estimates of an associated dual problem, together with the Galerkin orthogonality of the finite element method. Based on this a posteriori bound, we design and implement the corresponding adaptive algorithm to ensure global control of the error with respect to a user-defined tolerance.

     

The Numerical Simulation of Space-Time Variable Fractional Order Diffusion Equation

Many physical processes appear to exhibit fractional order behavior that may vary with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. Numerical methods and analysis of stability and convergence of numerical scheme for the variable fractional order partial differential equations are quite limited and difficult to derive. This motivates us to develop efficient numerical methods as well as stability and convergence of the implicit numerical methods for the space-time variable fractional order diffusion equation on a finite domain. It is worth mentioning that here we use the Coimbra-definition variable time fractional derivative which is more efficient from the numerical standpoint and is preferable for modeling dynamical systems. An implicit Euler approximation is proposed and then the stability and convergence of the numerical scheme are investigated. Finally, numerical examples are provided to show that the implicit Euler approximation is computationally efficient.     

The Fixity of Groups of Prime-Power Order

It is well known that the only groups of prime-power order whichcan act fixed point freely on a complex linear space are thecyclic or generalised quaternion groups. Given a positive integerf and a prime p exceeding f, we determine the p-groups whichhave a faithful complex representation such that the dimensionsof the spaces of fixed points of non-trivial elements are atmost f.