We propose a spectral viscosity method to approximate the two-dimensional Euler equations with rough initial data and prove that the method converges to a weak solution for a large class of initial data, including when the initial vorticity is in the so-called Delort class, i.e., it is a sum of a signed measure and an integrable function. This provides the first convergence proof for a numerical method approximating the Euler equations with such rough initial data and closes the gap between the available existence theory and rigorous convergence results for numerical methods. We also present numerical experiments, including computations of vortex sheets and confined eddies, to illustrate the proposed method.

我们提出了一种频谱粘度方法，用粗糙的初始数据来近似二维Euler方程，并证明了该方法对于大类初始数据（包括初始涡度在所谓的Delort类中）收敛到一个弱解，即，它是有符号测度和可积分函数的和。这为用这样粗略的初始数据逼近Euler方程的数值方法提供了第一个收敛性证明，并缩小了可用的存在理论与数值方法的严格收敛结果之间的差距。我们还提出了数值实验，包括涡旋片和密闭涡的计算，以说明所提出的方法。