Motion of a vortex sheet on a sphere with pole vortices

We cons i der the motion of a vortex sheet on the surface of a unit sphere in the presence of point vortices xed on north and south poles.Analytic and numerical research revealed that a vortex sheet in two-dimensional space has the following three properties.First,the vortex sheet is linearly unstable due to Kelvin-Helmholtz instability.Second,the curvature of the vortex sheet diverges in nite time.Last,the vortex sheet evolves into a rolling-up doubly branched spiral,when the equation of motion is regularized by the vortex method.The purpose of this article is to investigate how the curvature of the sphere and the presence of the pole vortices

Mechanical Quadrature Method and Splitting Extrapolation for Solving Dirichlet Boundary Integral Equation of Helmholtz Equation on Polygons

We study the numerical solution of Helmholtz equation with Dirichlet boundary condition. Based on the potential theory, the problem can be converted into a boundary integral equation. We propose the mechanical quadrature method (MQM) using specific quadrature rule to deal with weakly singular integrals. Denote by hm the mesh width of a curved edge Γm  (m=1,…,d) of polygons. Then, the multivariate asymptotic error expansion of MQM accompanied with O(hm3) for all mesh widths hm is obtained. Hence, once discrete equations with coarse meshes are solved in parallel, the higher accuracy order of numerical approximations can be at least O(hmax⁡5) by splitting extrapolation algorithm (SEA). A numerical example is provided to support our theoretical analysis