Volterra integro-differential equations with time-dependent delay arguments (DVIDEs) can provide us with realistic models of many real-world phenomena. Delayed LoktaVolterra predator-prey systems arise in Ecology and are well-known examples of DVIDEs first introduced by Volterra in 1928. We investigate the numerical solution of systems of DVIDEs using an adaptive stepsize selection strategy. We...

We consider variable stepsize time approximations of holomorphic semigroups on general Banach spaces. For strongly A(0)-acceptable rational functions a general stability theorem is proved, which does not impose any constraint on the ratios between stepsizes.

The numerical integration of reversible dynamical systems is considered. A backward analysis for variable stepsize one-step methods is developed, and it is shown that the numerical solution of a symmetric one-step method, implemented with a reversible stepsize strategy, is formally equal to the exact solution of a perturbed differential equation, which again is reversible. This explains geometr...

This paper is concerned with the stability of rational one-step approximations of C0 semigroups. Particular emphasis is laid on long-term stability bounds. The analysis is based on a general Banach space framework and allows variable stepsize sequences. Under reasonable assumptions on the stepsize sequence, asymptotic stability bounds for general C0 semigroups are derived. The bounds are typica...

The development of a Sundman-type time-transformation for reversible variable stepsize integration of few-body problems is discussed. While a time-transformation based on minimum particle separation is suitable if the collisions only occur pairwise and isolated in time, the control of stepsize is typically much more diicult for a three-body close approach. Nonetheless, we nd that a suitable cho...

Time-stepping methods that guarantee to avoid spurious fixed points are said to be regular. For fixed stepsize Runge-Kutta formulas, this concept has been well studied. Here, the theory of regularity is extended to the case of embedded Runge-Kutta pairs used in variable stepsize mode with local error control. First, the limiting case of a zero error tolerance is considered. A recursive regulari...

The development of a Sundman-type time-transformation for reversible variable stepsize integration of few-body problems is discussed. While a time-transformation based on minimum particle separation is suitable if the collisions only occur pairwise and isolated in time, the control of stepsize is typically much more diicult for a three-body close approach. Nonetheless, we nd that a suitable cho...

We assess the performance of an exponential integrator for advancing stiff, semidiscrete formulations of the unsaturated Richards equation in time. The scheme is of second order and explicit in nature but requires the action of the matrix function φ(A) = A−1(eA − I) on a suitability defined vector v at each time step. When the matrix A is large and sparse, φ(A)v can be approximated by Krylov su...

This paper presents the convergence analysis result of the discrete cosine transform-least-mean-square (DCT-LMS) adaptive "ltering algorithm which is based on a well-known interpretation of the variable stepsize algorithm. The time-varying stepsize of the DCT-LMS algorithm is implemented by the modi"ed power estimator to redistribute the spread power after the DCT. The performance analysis is c...