We present an accurate and efficient boundary integral (BI) method for simulating the deformation of drops and bubbles in Stokes flow with soluble surfactant. Soluble surfactant advects and diffuses in bulk fluids while adsorbing and desorbing at interfaces. Since the fluid velocity is coupled to the surfactant concentration, the advection-diffusion equation governing the bulk surfactant concentration $C$ is nonlinear, precluding the Green's function formulation necessary for a BI method. However, in the physically representative large P\'eclet number limit, an analytical reduction of the surfactant dynamics permits a Green's function formulation for $C$ as an Abel-type time-convolution integral at each Lagrangian interface point. A challenge in developing a practical numerical method based on this formulation is the fast evaluation of the time convolution, since the kernel depends on the time history of quantities at the interface, which is only found during the time-stepping process. To address this, we develop a novel, causal version of the Fast Multipole Method that reduces the computational cost from $O(P^2)$ for direct evaluation of the time convolution to $O(P \log_2^2 P)$ per surface grid point, where $P$ is the number of time steps. In the bulk phase, the resulting method is mesh-free and provides an accurate solution to the fully coupled moving interface problem with soluble surfactant. The approach extends naturally to a broader class of advection-diffusion problems in the high P\'eclet number regime.

翻译：我们提出了一种精确高效的边界积分法，用于模拟含可溶性表面活性剂的斯托克斯流中液滴与气泡的形变过程。可溶性表面活性剂在体相流体中通过对流-扩散输运，同时在界面处发生吸附与脱附。由于流体速度与表面活性剂浓度相互耦合，控制体相表面活性剂浓度$C$的对流-扩散方程呈现非线性特征，导致边界积分法所必需的格林函数表述无法直接建立。然而，在具有物理代表性的高佩克莱特数极限下，通过对表面活性剂动力学进行解析简化，可将$C$的格林函数表述转化为每个拉格朗日界面点上的阿贝尔型时间卷积积分形式。基于该表述构建实用数值方法的核心挑战在于时间卷积的快速计算——因其积分核依赖于界面物理量的时间历程，而这些量仅在时间步进过程中逐步获得。为此，我们发展了一种新颖的因果型快速多极子算法，将每个表面网格点的时间卷积计算复杂度从直接计算的$O(P^2)$降低至$O(P \log_2^2 P)$，其中$P$为时间步数。在体相区域，该方法保持无网格特性，能够精确求解含可溶性表面活性剂的完全耦合运动界面问题。该研究思路可自然推广至高佩克莱特数条件下更广泛的对流-扩散问题类别。