High-Order and Efficient Methods for the Vorticity Formulation of the Euler Equations

Abstract

In this work, the authors develop new methods for the accurate and efficient solution of the two-dimensional, incompressible Euler equations in the vorticity form. Here, the velocity is recovered directly from the Biot–Savart relation with vorticity, and the vorticity is evolved through its transport equation. Using a generalized Poisson summation formula, the full asymptotic error expansion is constructed for the second-order point vortex approximation to the Biot–Savart integral over a rectangular grid. The expansion is in powers of $h^2 $, and its coefficients depend linearly upon only local derivatives of the vorticity. In particular, the second-order term depends only upon the vorticity gradient. Except at second-order, the coefficients also involve rapidly convergent, two-dimensional lattice sums. At second-order, the sum is conditionally convergent, but can be calculated easily and rapidly. Therefore, we can remove the second-order term explicitly from the point vortex approximation to obtain a fourth-order discretization. In the special case of a square grid, the second-order error term is orthogonal to the vorticity gradient. The convective derivative of vorticity is thus calculated to fourth order using the unmodified point vortex approximation and automatically yields a fourth-order evolution. These methods have been implemented. For the vorticity transport equation, the point vortex sums are evaluated very rapidly using the fast Fourier transform (FF1) algorithm, and vorticity gradients are approximated using high-order difference methods. With high resolution, the authors solve numerically the roll-up of a thin layer of vorticity through the Kelvin–Helmholtz instability, and the interaction of two oppositely signed vortices driven together under an external strain flow. In the latter case, a simple time dependence of the grid is introduced to maintain resolution of the flow.