Regularization of contour dynamical algorithms. I. tangential regularization

Abstract

Contour dynamical methods are being applied to a variety of inviscid incompressible flows in two dimensions. These generalizations of the “waterbag” method provide simplified models for following the evolution of contours x (J) that separate regions of constant density which are the sources of the flow. The inviscid evolution of contour j, x̂ t (j) is usually an area-preserving map. For physically unstable problems, a piecewise-constant initial condition may result in an ill-posed problem. That is, contours may rapidly grow in perimeter and/or develop singularities and numerically induced small-scale structures in a finite time. To avoid such problems and model realistic weakly dissipative or weakly dispersive flows, contour regularization procedures are required. Dissipative and dispersive tangential regularization procedures for one contour are introduced. A special case of the former, namely ẋ t = μx ss , corresponds in lowest order to a linear diffusion operator in two dimensions. The contour is parameterized with arc length using cubic splines and an adaptive curvature controlled node adjustment algorithm is used. A modified Crank-Nicolson method is used to solve the discrete representation of the full system, x t = x̂ t +μx ss . Numerical results are given for the evolution of initially elliptical shapes according to prescribed area-preserving maps. The numerical results for area evolution agree with analytical results.