Contour Dynamics with Symplectic Time Integration

Abstract

In this paper we consider the time evolution of vortices simulated by the method of contour dynamics. Special attention is being paid to the Hamiltonian character of the governing equations and in particular to the conservational properties of numerical time integration for them. We assess symplectic and non-symplectic schemes. For the former methods, we give an implementation which is both efficient and yet effectively explicit. A number of numerical examples sustain the analysis and demonstrate the usefulness of the approach.

Collapse interactions of finite-sized two-dimensional vortices

“…This technique is not discussed here; for details see Dritschel 11,12 , Vosbeek and Mattheij. 14 Integrating the velocities over a time step ⌬t yields the positions of the boundaries after time step ⌬t and thus the evolution of the regions of uniform vorticity can be calculated. The time integration is performed using a secondorder, symplectic, mid-point rule ͑see Sanz-Serna and Calvo 15 ͒.…”

Acceleration of Contour Dynamics Simulations with a Hierarchical-Element Method

“…The initial configuration consists of three concentric, slightly elliptically disturbed, contours (aspect ratio is equal to ½¼¼ ). The outer ring has negative vorticity, while the core (consisting of the area enclosed by the second contour) has positive vorticity (see also the paper by Vosbeek and Mattheij [22]). Due to the elliptical disturbance, the monopole deforms and becomes a tripole while the core is becoming more elliptical.…”

“…Furthermore, it can be derived that the velocity field Ù´Ü Ø µ, anywhere in the flow, and in particular on the contours Ñ where ´Ü Ø µ is discontinuous, can be determined by the computation of contour integrals [7,8,20,22,23]:…”

“…Between two subsequent nodes on a contour, linear interpolation is used to determine the contour integrals in (8). The adding and removal of nodes is based on the local curvature of the contours, minimum and maximum distance between two successive nodes and quasi-uniformity of the distribution of the nodes [20,22].…”

“…The time integration is carried out using second order (symplectic) midpoint rule [19]. The reason for choosing this scheme is that it conserves quantities like the area and circulation of the regions of uniform vorticity better than ordinary integration methods [20,22].…”