Convergence results for the MATLAB ode23 routine

We prove convergence results on finite time intervals, as the user-defined tolerance τ→0, for a class of adaptive timestepping ODE solvers that includes the ode23 routine supplied in MATLAB Version 4.2. In contrast to existing theories, these convergence results hold with error constants that are uniform in the neighbourhood of equilibria; such uniformity is crucial for the derivation of results concerning the numerical approximation of dynamical systems. For linear problems the error estimates are uniform on compact sets of initial data. The analysis relies upon the identification of explicit embedded Runge-Kutta pairs for which all but the leading order terms of the expansion of the local error estimate areO(∥f(u∥)²). This work was partially supported by NSF Grant DMS-95-04879.     

Dissipativity of Runge-Kutta methods in Hilbert spaces

This paper concerns the discretization by Runge-Kutta methods of the initial value problemu _t =f(u), under the dissipative structural condition that there exist α≥0, β>0, such thatf:W→H, ℜe, ∀w∈W, for complex Hilbert spacesW⊆H. It is shown that strong A-stability is necessary to ensure the dissipativity of the method, whilst algebraic stability plus |R(∞)|<1 is a sufficient condition in the case of DJ-irreducible methods.     

A finite difference analysis of a streamline diffusion method on a Shishkin mesh

We consider streamline diffusion finite element methods applied to a singularly perturbed convection–diffusion two‐point boundary value problem whose solution has a single boundary layer. To analyse the convergence of these methods, we rewrite them as finite difference schemes. We first consider arbitrary meshes, then, in analysing the scheme on a Shishkin mesh, we consider two formulations on the fine part of the mesh: the usual streamline diffusion upwinding and the standard Galerkin method. The error estimates are given in the discrete L _∞ norm; in particular we give the first analysis that shows precisely how the error depends on the user-chosen parameter _τ0 specifying the mesh. When _τ0 is too small, the error becomes O(1), but for _τ0 above a certain threshold value, the error is small and increases either linearly or quadratically as a function of . Numerical tests support our theoretical results. This revised version was published online in August 2006 with corrections to the Cover Date.     

A spectral Galerkin approximation of the Orr-Sommerfeld eigenvalue problem in a semi-infinite domain

Summary The convergence of a Galerkin approximation of the Orr-Sommerfeld eigenvalue problem, which is defined in a semi-infinite domain, is studied theoretically. In case the system of trial functions is based on a composite of Jacobi polynomials and an exponential transform of the semi-infinite domain, the error of the Galerkin approximation is estimated in terms of the transformation parametera and the numberN of trial functions. Finite or infinite-order convergence of the spectral Galerkin method is obtained depending on how the transformation parameter is chosen. If the transformation parameter is fixed, then convergence is of finite order only. However, ifa is varied proportional to 1/N with an exponent 0<<1, then the approximate eigenvalue converges faster than any finite power of 1/N asN. Some numerical examles are given.     

Asymptotics for transient and stationary probabilities for finite and infinite buffer discrete time queues

Consider a discrete time queue with i.i.d. arrivals (see the generalisation below) and a single server with a buffer length m. Let τm be the first time an overflow occurs. We obtain asymptotic rate of growth of moments and distributions of τm as m → ∞. We also show that under general conditions, the overflow epochs converge to a compound Poisson process. Furthermore, we show that the results for the overflow epochs are qualitatively as well as quantitatively different from the excursion process of an infinite buffer queue studied in continuous time in the literature. Asymptotic results for several other characteristics of the loss process are also studied, e.g., exponential decay of the probability of no loss (for a fixed buffer length) in time [0,η], η → ∞, total number of packets lost in [0, η, maximum run of loss states in [0, η]. We also study tails of stationary distributions. All results extend to the multiserver case and most to a Markov modulated arrival process. This revised version was published online in June 2006 with corrections to the Cover Date.     

Eigenvalue distributions from impacts on a ring

We consider the collision dynamics produced by three beads with masses (m ₁, m ₂, m ₃) sliding without friction on a ring, where the masses are scaled so that m ₁ = 1/ɛ, m ₂ = 1, m ₃ = 1 − ɛ, for 0 ⩽ ɛ ⩾ 1. The singular limits ɛ = 0 and ɛ = 1 correspond to two equal mass beads colliding on the ring with a wall, and without a wall respectively. In both these cases, all solutions are periodic and the eigenvalue distributions (around the unit circle) associated with the products of collision matrices are discrete. We then numerically examine the regime which parametrically connects these two states, i.e. 0 < ɛ < 1, and show that the eigenvalue distribution is generically uniform around the unit circle, which implies that the dynamics are no longer periodic. By a sequence of careful numerical experiments, we characterize how the uniform spectrum collapses from continuous to discrete in the two singular limits ɛ → 0 and ɛ → 1 for an ensemble of initial velocities sampled uniformly on a fixed energy surface. For the limit ɛ → 0, the distribution forms Gaussian peaks around the discrete limiting values ± 1, ± i, with variances that scale in power law form as σ ² ∼ αɛ ^β. By contrast, the convergence in the limit ɛ → 1 to the discrete values ±1 is shown to follow a logarithmic power-law σ ² ∼ log(ɛ ^β).     

Non-relativistic global limits of entropy solutions to the isentropic relativistic Euler equations

We consider in this paper the relativistic Euler equations in isentropic fluids with the equation of state p = κ²ρ, where κ, the sound speed, is a constant less than the speed of light c. We discuss the convergence of the entropy solutions as c→∞. The analysis is based on the geometric properties of nonlinear wave curves and the Glimm’s method.     

Convergence and stability of implicit runge-kutta methods for systems with multiplicative noise

A class ofimplicit Runge-Kutta schemes for stochastic differential equations affected bymultiplicative Gaussian white noise is shown to be optimal with respect to global order of convergence in quadratic mean. A test equation is proposed in order to investigate the stability of discretization methods for systems of this kind. Herestability is intended in a truly probabilistic sense, as opposed to the recently introduced extension of A-stability to the stochastic context, given for systems with additive noise. Stability regions for the optimal class are also given.Partially supported by the Italian Consiglio Nazionale delle Ricerche.     

Martingale transforms and L p -norm estimates of Riesz transforms on complete Riemannian manifolds

Under the condition that the Bakry–Emery Ricci curvature is bounded from below, we prove a probabilistic representation formula of the Riesz transforms associated with a symmetric diffusion operator on a complete Riemannian manifold. Using the Burkholder sharp L ^p -inequality for martingale transforms, we obtain an explicit and dimension-free upper bound of the L ^p -norm of the Riesz transforms on such complete Riemannian manifolds for all 1 < p < ∞. In the Euclidean and the Gaussian cases, our upper bound is asymptotically sharp when p→ 1 and when p→ ∞. Research partially supported by a Delegation in CNRS at the University of Paris-Sud during the 2005–2006 academic year.     

Stiff Oscillatory Systems, Delta Jumps and White Noise

Two model problems for stiff oscillatory systems are introduced. Both comprise a linear superposition of harmonic oscillators used as a forcing term for a scalar ODE. In the first case the initial conditions are chosen so that the forcing term approximates a delta function as and in the second case so that it approximates white noise. In both cases the fastest natural frequency of the oscillators is <e6>OM</e6>(N). The model problems are integrated numerically in the stiff regime where the time-step satisfies The convergence of the algorithms is studied in this case in the limit and For the white noise problem both strong and weak convergence are considered. Order reduction phenomena are observed numerically and proved theoretically. August 25, 1999. Final version received: May 3, 2000.