In this paper, an adaptive error-controlled hybrid fast solver that combines both $\mathcal{O}(N)$ and $\mathcal{O}(N\log N)$schemes is proposed. For a given accuracy, the adaptive solver is used in the context of regularized vortex methods to optimize the speed of the velocity and vortex stretching calculation. This is accomplished by introducing criteria for cell division in building of the tree, conversion of multipole to local expansion in the downward pass, stopping of the downward pass and choosing between direct and fast summation to compute the vector fields. These criteria are based on key parameters ($p,n_F,n_T,d_{\sigma }$) which take into account the elements distribution, choice of the regularization function, and the computer architecture. The proposed solver automatically adapts to the evolving flow-field by periodically updating the optimal values of these parameters to maximize the speed, while meeting the accuracy constraints, by balancing far and near-field calculations. Performance of the proposed scheme is investigated in terms of the dependence of cost and accuracy on the various controlling parameters. The evolution of the optimal values of these parameters along with the associated computational savings are presented for the case of collision of two vortex rings over a reasonable time span.

本文提出了一种自适应误差控制的混合快速求解器，它结合了两者$\mathcal{○}(ñ)$和$\mathcal{○}(ñ\mathrm{日志}ñ)$提出了方案。对于给定的精度，自适应求解器用于正则化涡流方法的上下文中，以优化速度和涡流拉伸计算的速度。这是通过在树的构建中引入细胞分裂标准、在向下传递中将多极子转换为局部扩展、停止向下传递以及在直接和快速求和之间进行选择来计算矢量场来实现的。这些标准基于关键参数（$p,n_F,n_{吨},d_{\sigma }$) 其中考虑了元素分布、正则化函数的选择和计算机体系结构。所提出的求解器通过定期更新这些参数的最优值以最大化速度，同时通过平衡远场和近场计算来自动适应不断变化的流场，同时满足精度约束。根据成本和精度对各种控制参数的依赖性，研究了所提出方案的性能。对于两个涡环在合理时间跨度内的碰撞情况，提出了这些参数的最佳值的演变以及相关的计算节省。