Numerical Methods for the 3-dimensional 2-body Problem in the Action-at-a-Distance Electrodynamics

We develop two numerical methods to solve the differential equations with deviating arguments for the motion of two charges in the action-at-a-distance electrodynamics. Our first method uses St\"urmer's extrapolation formula and assumes that a step of integration can be taken as a step of light ladder, which limits its use to shallow energies. The second method is an improvement of pre-existing iterative schemes, designed for stronger convergence and can be used at high-energies.Comment: 17 pages, 11 figure

A comparative study of ADI splitting methods for parabolic equations in two space dimensions

AbstractThe main purpose of the paper is a numerical comparison of three integration methods for semi-discrete parabolic partial differential equations in two space variables. Linear as well as nonlinear,equations are considered. The integration methods are the well-known ADI method of Peaceman and Rachford, a global extrapolation scheme of the classical ADI method to order four and a fourth order, four-step ADI splitting method

Exact numerical methods for a many-body Wannier Stark system

We present exact methods for the numerical integration of the Wannier-Stark system in a many-body scenario including two Bloch bands. Our ab initio approaches allow for the treatment of a few-body problem with bosonic statistics and strong interparticle interaction. The numerical implementation is based on the Lanczos algorithm for the diagonalization of large, but sparse symmetric Floquet matrices. We analyze the scheme efficiency in terms of the computational time, which is shown to scale polynomially with the size of the system. The numerically computed eigensystem is applied to the analysis of the Floquet Hamiltonian describing our problem. We show that this allows, for instance, for the efficient detection and characterization of avoided crossings and their statistical analysis. We finally compare the efficiency of our Lanczos diagonalization for computing the temporal evolution of our many-body system with an explicit fourth order Runge-Kutta integration. Both implementations heavily exploit efficient matrix-vector multiplication schemes. Our results should permit an extrapolation of the applicability of exact methods to increasing sizes of generic many-body quantum problems with bosonic statistics

Step size adjustment and extrapolation for time-stepping schemes in non-smooth dynamics

International audienceIn this paper we use step size adjustment and extrapolation methods to improve Moreau's time-stepping scheme for the numerical integration of non-smooth mechanical systems, i.e. systems with impact and friction. The scheme yields a system of inclusions, which is transformed into a system of projective equations. These equations are solved iteratively. Switching points are time instants for which the structure of the mechanical system changes, for example, time instants for which a sticking friction element begins to slide. We show how switching points can be localized and how these points can be resolved by choosing a minimal step size. In order to improve the integration of non-smooth systems in the smooth parts, we show how the time-stepping method can be used as a base integration scheme for extrapolation methods, which allow for an increase in the integration order. Switching points are processed by a small time step, while time intervals during which the structure of the system does not change are computed with a larger step size and improved integration order. The overall algorithm, which consists of a time-stepping module, an extrapolation module and a step size adjustment module, is discussed in detail and some examples are given

Solving Reaction-Diffusion and Advection Problems with Richardson Extrapolation

Richardson extrapolation is a simple but powerful computational tool to enhance the accuracy of time integration methods. In the past years a few theoretical and partly practical works have been presented on this method. Detailed numerical applications of this method, however, are rarely found in the literature. Therefore, it is worth investigating whether this promising technique lives up to the expectations also in practice. In this paper we investigate the efficiency of the Richardson method in one-dimensional numerical (reaction-diffusion) problems

A comparison of high-order time integrators for thermal convection in rotating spherical shells

A numerical study of several time integration methods for solving the threedimensional Boussinesq thermal convection equations in rotating spherical shells is presented. Implicit and semi-implicit time integration techniques based on backward differentiation and extrapolation formulae are considered. The use of Krylov techniques allows the implicit treatment of the Coriolis term with low storage requirements. The codes are validated with a known benchmark, and their efficiency is studied. The results show that the use of high order methods, especially those with time step and order control, increase the efficiency of the time integration, and allows to obtain more accurate solutions.Postprint (author’s final draft

Nonlinear force-free modeling of the solar coronal magnetic field

The coronal magnetic field is an important quantity because the magnetic field dominates the structure of the solar corona. Unfortunately direct measurements of coronal magnetic fields are usually not available. The photospheric magnetic field is measured routinely with vector magnetographs. These photospheric measurements are extrapolated into the solar corona. The extrapolated coronal magnetic field depends on assumptions regarding the coronal plasma, e.g. force-freeness. Force-free means that all non-magnetic forces like pressure gradients and gravity are neglected. This approach is well justified in the solar corona due to the low plasma beta. One has to take care, however, about ambiguities, noise and non-magnetic forces in the photosphere, where the magnetic field vector is measured. Here we review different numerical methods for a nonlinear force-free coronal magnetic field extrapolation: Grad-Rubin codes, upward integration method, MHD-relaxation, optimization and the boundary element approach. We briefly discuss the main features of the different methods and concentrate mainly on recently developed new codes.Comment: 33 pages, 3 figures, Review articl

An improved unified solver for compressible and incompressible fluids involving free surfaces. II. Multi-time-step integration and applications

An improved numerical solver for the unified solution of compressible and incompressible fluids involving interfaces is proposed. The present method is based on the CIP-CUP (Cubic Interpolated Propagation / Combined, Unified Procedure) method, which is a pressure-based semi-implicit solver for the Euler equations of fluid flows. In Part I of this series of articles [M. Ida, Comput. Phys. Commun. 132 (2000) 44], we proposed an improved scheme for the convection terms in the equations, which allowed us discontinuous descriptions of the density interface by replacing the cubic interpolation function used in the CIP scheme with a quadratic extrapolation function only around the interface. In this paper, as Part II of this series, the multi-time-step integration technique is adapted to the CIP-CUP integration. Because the CIP-CUP treats different-nature components in the fluid equations separately, the adaptation of the technique is straightforward. This modification allows us flexible determinations of the time interval, which results in an efficient and accurate integration. Furthermore, some additional discussion on our methods is presented. Finally, the application results to composite flow problems such as compressible and incompressible Kelvin-Helmholtz instabilities and the dynamics of two acoustically coupled deformable bubbles in a viscous liquid are provided.Comment: 34 pages, 13 figures, elsart; Typo in Eq.25 corrected; Publishe