Some geometrical algorithms for generation of field lines of a vector field?designed for high accuracy with calculations at only a few field points

Abstract

Toward the goal of efficiently computing field lines for molecular vector fields, where each field-point calculation is computationally intensive, a few appropriate algorithms for calculating vector field lines are presented and compared for various representative applications. Among these algorithms are the first-order tangent-line method (TL), the fourth-order Runge-Kutta method (RK4), the infinite-order Bulirsch-Stoer method (BS), the second-order Taylor's series method (TS2), and the second-order “curvature-following” method (CF). The TL and the RK4 are well known. The TS2 and the CF are new. The RK4, The TS2, and the CF are appropriate for obtaining high accuracy with few field-point calculations. The TL is definitely not appropriate for this purpose, and the BS is so appropriate only at the highest level of required accuracy. The CF uses the value of the vector field and its gradient at the given field point in order to locate the center of curvature of the field line at that point and, thereby, to extrapolate the field line, as an arc of a circle, to the next field point. The TS2 uses the same information, but extrapolates the field line as a segment of a parabola whose vertex is at the field point. The BS is an infinite-order extrapolation on successively finer scale iterations of the lowest-order Runge-Kutta method. All of these methods are compared for the velocity field of a rotating disk, for the vector field of a point dipole, and for electric field of a high-speed orbiting charged particle. For all of these fields, the field lines are exactly expressible in analytic form, so the absolute errors of these different algorithms can be appraised. As several of these methods allow a rather large step size, it was found appropriate to use a continuous-(geometrical)-curvature interpolation scheme to interpolate between the field points on the generated field lines. One prefers this scheme to the use of cubic splines when the physical fields should have their “geometrical pictures” (even under approximation) invariant to an arbitrary change of the coordinate system used in the calculation. These methods have also been used to generate field lines for the approximate and the exact fields of a half-wave antenna. In this case, one sees very large differences in the structure of the field lines of the approximate and the exact fields, in the near-field region, and also sees the manner in which each of these differences diminishes to zero as the field point approaches the far-field region; features that would have been very hard to observe by purely analytical methods. It is hoped that these methods for field line generation might have application for the field lines of the gradient of the molecular density (as in Bader's theory of atoms in molecules) and for the field lines of the electric fields of nucleic acid molecules.