High-order and efficient methods for the vorticity formulation of the Euler equations

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 fou...