325 research outputs found
Linear iterative solvers for implicit ODE methods
The numerical solution of stiff initial value problems, which lead to the problem of solving large systems of mildly nonlinear equations are considered. For many problems derived from engineering and science, a solution is possible only with methods derived from iterative linear equation solvers. A common approach to solving the nonlinear equations is to employ an approximate solution obtained from an explicit method. The error is examined to determine how it is distributed among the stiff and non-stiff components, which bears on the choice of an iterative method. The conclusion is that error is (roughly) uniformly distributed, a fact that suggests the Chebyshev method (and the accompanying Manteuffel adaptive parameter algorithm). This method is described, also commenting on Richardson's method and its advantages for large problems. Richardson's method and the Chebyshev method with the Mantueffel algorithm are applied to the solution of the nonlinear equations by Newton's method
A class of symplectic integrators with adaptive timestep for separable Hamiltonian systems
Symplectic integration algorithms are well-suited for long-term integrations
of Hamiltonian systems because they preserve the geometric structure of the
Hamiltonian flow. However, this desirable property is generally lost when
adaptive timestep control is added to a symplectic integrator. We describe an
adaptive-timestep symplectic integrator that can be used if the Hamiltonian is
the sum of kinetic and potential energy components and the required timestep
depends only on the potential energy (e.g. test-particle integrations in fixed
potentials). In particular, we describe an explicit, reversible, symplectic,
leapfrog integrator for a test particle in a near-Keplerian potential; this
integrator has timestep proportional to distance from the attracting mass and
has the remarkable property of integrating orbits in an inverse-square force
field with only "along-track" errors; i.e. the phase-space shape of a Keplerian
orbit is reproduced exactly, but the orbital period is in error by O(1/N^2),
where N is the number of steps per period.Comment: 24 pages, 3 figures, submitted to Astronomical Journal; minor errors
in equations and one figure correcte
A Hybrid N-body--Coagulation Code for Planet Formation
We describe a hybrid algorithm to calculate the formation of planets from an
initial ensemble of planetesimals. The algorithm uses a coagulation code to
treat the growth of planetesimals into oligarchs and explicit N-body
calculations to follow the evolution of oligarchs into planets. To validate the
N-body portion of the algorithm, we use a battery of tests in planetary
dynamics. Several complete calculations of terrestrial planet formation with
the hybrid code yield good agreement with previously published calculations.
These results demonstrate that the hybrid code provides an accurate treatment
of the evolution of planetesimals into planets.Comment: Astronomical Journal, accepted; 33 pages + 11 figure
Canonical Runge—Kutta—Nyström methods of orders five and six
AbstractIn this paper, we construct canonical explicit five-stage and seven-stage Runge—Kutta—Nyström methods of orders five and six, respectively, for Hamiltonian dynamical systems
On an asymptotic method for computing the modified energy for symplectic methods
We revisit an algorithm by Skeel et al. [5,16] for computing the modified, or shadow, energy associated with symplectic discretizations of Hamiltonian systems. We amend the algorithm to use Richardson extrapolation in order to obtain arbitrarily high order of accuracy. Error estimates show that the new method captures the exponentially small drift associated with such discretizations. Several numerical examples illustrate the theory
- …