An explicit time-stepping method is developed for adaptive solution of time-dependent partial differential equations with first order derivatives. The space is partitioned into blocks and the grid is refined and coarsened in these blocks. The equations are integrated in time by a Runge-KuttaFehlberg method. The local errors in space and time are estimated and the time and space steps are determ...

We consider canonical partitioned Runge-Kutta methods for separable Hamiltonians H = T(ß) + Viq) and canonical Runge-Kutta-Nyström methods for Hamiltonians of the form H = ^pTM~lp + Viq) with M a diagonal matrix. We show that for explicit methods there is great simplification in their structure. Canonical methods of orders one through four are constructed. Numerical experiments indicate the sui...

The parametric instability arising when ordinary differential equations (ODEs) are numerically integrated with Runge-Kutta-Nyström (RKN) methods with varying step sizes is investigated. Perturbation methods are used to quantify the critical step sizes associated with parametric instability. It is shown that there is no parametric instability for linear constant coefficient ODEs integrated with ...

This paper presents a family of Runge{Kutta type integration schemes of arbitrarily high order for di erential equations evolving on manifolds. We prove that any classical Runge{Kutta method can be turned into an invariant method of the same order on a general homogeneous manifold, and present a family of algorithms that are relatively simple to implement.

A fourth-order, implicit, low-dispersion, and low-dissipation Runge-Kutta scheme is introduced. The scheme is optimized for minimal dissipation and dispersion errors. High order accuracy is achieved with fewer stages than standard explicit Runge-Kutta schemes. The scheme is designed to be As table for highly stiff problems. Possible applications include wall-bounded flows with solid boundaries ...

In this paper, we construct canonical explicit 5-stage and 7-stage Runge-KuttaNyström methods of orders 5 and 6, respectively, for Hamiltonian dynamical systems.

The Multistage Differential Transform Method (MDTM) is employed to solve the model for HIV infection of CD4T cells. Comparing the numerical results to those obtained by the classical fourth order Runge-Kutta method showed the preciseness and efficacy of the multistep differential transform method. The study shows that the method is a powerful and promising tool for solving coupled systems of di...

We study numerical integrators that contract phase space volume even when the ODE does so at an arbitrarily small rate. This is done by a splitting into two-dimensional contractive systems. We prove a sufficient condition for Runge–Kutta methods to have the appropriate contraction property for these two-dimensional systems; the midpoint rule is an example.  2000 IMACS. Published by Elsevier Sc...

The gravitational anomalies are investigated for generalized Euclidean Taub-NUT metrics which admit hidden symmetries analogous to the Runge-Lenz vector of the Kepler-type problem. In order to evaluate the axial anomalies, the index of the Dirac operator for these metrics with the APS boundary condition is computed. The role of the Killing-Yano tensors is discussed for these two types of quantu...