A Two Step, Fourth Order, Nearly-Linear Method with Energy Preserving Properties

We introduce a family of fourth order two-step methods that preserve the energy function of canonical polynomial Hamiltonian systems. Each method in the family may be viewed as a correction of a linear two-step method, where the correction term is O(h^5) (h is the stepsize of integration). The key tools the new methods are based upon are the line integral associated with a conservative vector field (such as the one defined by a Hamiltonian dynamical system) and its discretization obtained by the aid of a quadrature formula. Energy conservation is equivalent to the requirement that the quadrature is exact, which turns out to be always the case in the event that the Hamiltonian function is a polynomial and the degree of precision of the quadrature formula is high enough. The non-polynomial case is also discussed and a number of test problems are finally presented in order to compare the behavior of the new methods to the theoretical results.Comment: 14 pages, 4 figures, 2 table

Numerical treatment of special second order ordinary differential equations: general and exponentially fitted methods

2010 - 2011The aim of this research is the construction and the analysis of new families of numerical methods for the integration of special second order Ordinary Differential Equations (ODEs). The modeling of continuous time dynamical systems using second order ODEs is widely used in many elds of applications, as celestial mechanics, seismology, molecular dynamics, or in the semidiscretisation of partial differential equations (which leads to high dimensional systems and stiffness). Although the numerical treatment of this problem has been widely discussed in the literature, the interest in this area is still vivid, because such equations generally exhibit typical problems (e.g. stiffness, metastability, periodicity, high oscillations), which must efficiently be overcome by using suitable numerical integrators. The purpose of this research is twofold: on the one hand to construct a general family of numerical methods for special second order ODEs of the type y00 = f(y(t)), in order to provide an unifying approach for the analysis of the properties of consistency, zero-stability and convergence; on the other hand to derive special purpose methods, that follow the oscillatory or periodic behaviour of the solution of the problem...[edited by author]X n. s

Adapted block hybrid method for the numerical solution of duffing equations and related problems

Problems of non-linear equations to model real-life phenomena have a long history in science and engineering. One of the popular of such non-linear equations is the Duffing equation. An adapted block hybrid numerical integrator that is dependent on a fixed frequency and fixed step length is proposed for the integration of Duffing equations. The stability and convergence of the method are demonstrated; its accuracy and efficiency are also established

Study of beam dynamics in NS-FFAG EMMA with dynamical map

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Dynamical maps for magnetic components are fundamental to studies of beam dynamics in accelerators. However, it is usually not possible to write down maps in closed form for anything other than simplified models of standard accelerator magnets. In the work presented here, the magnetic field is expressed in analytical form obtained from fitting Fourier series to a 3D numerical solution of Maxwell’s equations. Dynamical maps are computed for a particle moving through this field by applying a second order (with the paraxial approximation) explicit symplectic integrator. These techniques are used to study the beam dynamics in the first non-scaling FFAG ever built, EMMA, especially challenging regarding the validity of the paraxial approximation for the large excursion of particle trajectories. The EMMA lattice has four degrees of freedom (strength and transverse position of each of the two quadrupoles in each periodic cell). Dynamical maps, computed for a set of lattice configurations, may be efficiently used to predict the dynamics in any lattice configuration. We interpolate the coefficients of the generating function for the given configuration, ensuring the symplecticity of the solution. An optimisation routine uses this tool to look for a lattice defined by four constraints on the time of flight at different beam energies. This provides a way to determine the tuning of the lattice required to produce a desired variation of time of flight with energy, which is one of the key characteristics for beam acceleration in EMMA. These tools are then benchmarked against data from the recent EMMA commissioning