Contour dynamics with symplectic time integration

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 nonsymplectic 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

Acceleration of contour dynamics simulations with a hierarchical-element method

This paper presents a so-called hierarchical-element method that can be used to accelerate complex contour dynamics simulations. The method is based on a modified fast multipole method where the multipole approximations are replaced by Poisson integrals. In this paper, attention is being paid to the theoretical derivation of the method. Furthermore, numerical and implementation aspects are considered. Various numerical simulations show that the speed-up of the method is significant, while the accuracy of the results is not being influenced

Contour dynamics with non-uniform background vorticity

In this paper it is demonstrated how a contour dynamics method can be used to simulate the behaviour of vortices in the presence of non-uniform background vorticity in general, and on the gamma -plane in particular. For standard contour dynamics in case of zero, or uniform background vorticity, the initial continuous vorticity distributions of the vortices are replaced by appropriate piecewise-uniform distributions. Then, the evolution of the contours separating the several regions of uniform vorticity, are followed in time. In the case of non-uniform background vorticity, it is necessary to replace the sum of the (relative) vorticity of the vortices and the background vorticity by a piecewise-uniform distribution. This has several consequences for applying the method of contour dynamics, which are discussed in this paper. The resulting method is tested on some numerical examples. One of them is (qualitatively) compared with laboratory experiments carried out in a rotating tank